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								<link href=".\..\..\toc_tuto.xmcd" title="Links to Tutorials" popup="false">tutorials</link>
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								<c val="#800000">Solving Equations and Systems</c>
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				<phrase>solving equations and systems</phrase>
				<phrase>solving systems of equations</phrase>
				<phrase>systems of equations, solving</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Learn both numeric and symbolic methods for solving single equations and systems of equations. </p>
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				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<c val="#000">Solving Equations Symbolically</c>
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					<link href="#solve keyword" title="Links to section" popup="false">Using the Symbolic Equal Sign with the Keyword "solve"</link>
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							<link href="#Variable Solve" popup="false">Using the Symbolics Menu</link>
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					<link href="#comparing1" title="Links to section" popup="false">Comparing the Two Methods</link>
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					<c val="#000">Solving Equations Numerically</c>
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								<link href="#using root function" popup="false">Using the root Function</link>
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					<c val="#000">Systems of Equations</c>
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					<link href="#setting equal" title="Links to section" popup="false">Solving Two Equations by Setting Them Equal to One Another</link>
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					<link href="#solve block" title="Links to section" popup="false">Using a Solve Block</link>
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					<b>
						<c val="#008080">Tip</c>
					</b>: The exercises in this tutorial require you to work within the Worksheet window. To copy the contents into the Worksheet window, choose <b>Select All</b> from the <b>Edit</b> menu. Then, either drag or copy and paste all the regions into your Worksheet window. Finally, click in the Worksheet window to complete the tasks below. </p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Before you start, size the Resources window and your Worksheet window so they are side-by-side on your desktop. </p>
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				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving Equations Symbolically</p>
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				<phrase>solve for variable</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When you solve an equation symbolically, you see the results in terms of variables and mathematical symbols. Seeing results in symbolic form can often give more insight into a problem than simply looking at numerical results. You can easily convert symbolic results to numerical ones as needed.<sp count="2"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are two ways to solve an equation symbolically:</p>
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				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
					<b>
						<c val="#008080">•<sp count="3"/>
						</c>
					</b>Use the symbolic equal sign (<f family="Symbol" charset="2">®) </f>with the keyword "solve." We<sp count="2"/>recommend this method because the results are "live" and update with changes you make above the equation in the worksheet. </p>
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit"/>
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
					<b>
						<c val="#008080">•<sp count="3"/>
						</c>
					</b>
					<f family="Times New Roman" charset="0" size="12">Select </f>
					<f family="Times New Roman" charset="0" size="12">
						<b>Variable &gt; Solve</b>
					</f>
					<f family="Times New Roman" charset="0" size="12"> from the </f>
					<b>
						<f family="Times New Roman" charset="0" size="12">Symbolics</f>
					</b>
					<f family="Times New Roman" charset="0" size="12"> menu. In this case, the results </f>
					<f family="Times New Roman" charset="0" size="12">are not automatically updated.</f>
				</p>
			</text>
		</region>
		<region region-id="1631" left="6" top="704.25" width="370.5" height="12" align-x="6" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="solve keyword">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using the Symbolic Equal Sign with the Keyword "solve"</p>
			</text>
			<indexes>
				<phrase>solve keyword</phrase>
			</indexes>
		</region>
		<region region-id="1380" left="12" top="726.75" width="341.25" height="42.75" align-x="12" align-y="738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>In your worksheet window, type the equation using the Boolean equal sign. To do so, type <b>[Ctrl] [</b>
					<inlineAttr font-weight="normal">=</inlineAttr>
					<b>]</b> or click the Equal To button on the <b>Boolean</b> toolbar.</p>
			</text>
		</region>
		<region region-id="1535" left="24" top="780" width="63.75" height="34.5" align-x="24" align-y="780" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag="">
			<picture>
				<png item-idref="4" display-width="62.25" display-height="33"/>
			</picture>
			<rendering item-idref="5"/>
		</region>
		<region region-id="1381" left="12" top="822.75" width="341.25" height="28.5" align-x="12" align-y="834" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<sp count="2"/>Click anywhere in the equation and press <b>[Ctrl] [Shift] [.]</b> to insert a placeholder followed by the symbolic equal sign. </p>
			</text>
		</region>
		<region region-id="1625" left="30" top="864" width="90" height="24" align-x="30" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<picture>
				<png item-idref="6" display-width="88.5" display-height="22.5"/>
			</picture>
			<rendering item-idref="7"/>
		</region>
		<region region-id="1383" left="12" top="900.75" width="341.25" height="28.5" align-x="12" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">3.</c>
					</b>
					<sp count="2"/>Type the keyword "solve" in the placeholder and press <b>[Enter]</b> to see the result displayed.</p>
			</text>
		</region>
		<region region-id="1542" left="30" top="944.25" width="136.5" height="36.75" align-x="126.75" align-y="966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:pow/>
								<ml:id xml:space="preserve">x</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
							<ml:real>4</ml:real>
						</ml:apply>
						<ml:real>21</ml:real>
					</ml:apply>
					<ml:command>
						<ml:id xml:space="preserve">solve</ml:id>
					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:real>-5</ml:real>
							<ml:real>5</ml:real>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="8"/>
		</region>
		<region region-id="1541" left="6" top="996.75" width="334.5" height="42.75" align-x="6" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<c val="#008080">Note: </c>
					<c val="#000">If the equation contains more than one variable, you must also type a comma after "solve" and then type the variable you want to solve for in the placeholder that appears. </c>
				</p>
			</text>
		</region>
		<region region-id="1430" left="0" top="1050" width="6000" height="6" align-x="0" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="814" left="6" top="1070.25" width="300.75" height="12" align-x="6" align-y="1080" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="Variable Solve">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Using the Symbolics Menu</p>
			</text>
		</region>
		<region region-id="1454" left="6" top="1092.75" width="329.25" height="42.75" align-x="24" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Another way to evaluate an expression symbolically is to use the <b>Symbolics</b> menu. However, in this case results are not automatically updated when you make changes elsewhere in the worksheet. </p>
			</text>
		</region>
		<region region-id="815" left="6" top="1152.75" width="341.25" height="42.75" align-x="6" align-y="1164" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>In your Worksheet window, type the equation you want to solve using the Boolean equal sign. To do so, type <b>[Ctrl] [=]</b> or click the Equal To button on the <b>Boolean</b> toolbar.</p>
			</text>
			<indexes>
				<phrase>Boolean equal sign</phrase>
			</indexes>
		</region>
		<region region-id="816" left="6" top="1206.75" width="272.25" height="28.5" align-x="6" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<sp count="2"/>Select just the variable you want to solve for by clicking on it. <b>Note:</b> Do not select the entire expression.</p>
			</text>
		</region>
		<region region-id="817" left="6" top="1248.75" width="263.25" height="14.25" align-x="6" align-y="1260" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">3.</c>
					</b>
					<sp count="2"/>Choose <b>Variable &gt; Solve</b> from the <b>Symbolics</b> menu. </p>
			</text>
		</region>
		<region region-id="1246" left="6" top="1286.25" width="50.25" height="12" align-x="24.75" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
			</text>
		</region>
		<region region-id="1097" left="18" top="1308" width="67.5" height="35.25" align-x="18" align-y="1308" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<picture>
				<png item-idref="9" display-width="66" display-height="33.75"/>
			</picture>
			<rendering item-idref="10"/>
		</region>
		<region region-id="1098" left="108" top="1308.75" width="138" height="42.75" align-x="108" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal">Click the x and then select </inlineAttr>
					</f>
					<f family="Times New Roman" charset="0">
						<b>Variable</b>
					</f>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal">
							<sp/>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0">
						<b>&gt; </b>
					</f>
					<f family="Times New Roman" charset="0">
						<b>Solve</b>
					</f>
					<inlineAttr font-weight="normal">
						<f family="Times New Roman" charset="0"> from the </f>
					</inlineAttr>
					<f family="Times New Roman" charset="0">
						<b>Symbolics</b>
					</f>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal"> menu. </inlineAttr>
					</f>
				</p>
			</text>
		</region>
		<region region-id="1099" left="24" top="1351.5" width="57.75" height="21" align-x="60.75" align-y="1368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:pow/>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:real>2</ml:real>
						</ml:apply>
						<ml:real>4</ml:real>
					</ml:apply>
					<ml:real>21</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="11"/>
		</region>
		<region region-id="1239" left="30" top="1394.25" width="30" height="36.75" align-x="44.25" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:matrix rows="2" cols="1" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:real>5</ml:real>
					<ml:real>-5</ml:real>
				</ml:matrix>
			</math>
			<rendering item-idref="12"/>
		</region>
		<region region-id="1421" left="6" top="1446.75" width="243" height="14.25" align-x="6" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Therefore both 5 and -5 are solutions to the equation.</p>
			</text>
		</region>
		<region region-id="1420" left="6" top="1478.25" width="174.75" height="12" align-x="6" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="comparing1">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Comparing the Two Methods</p>
			</text>
			<indexes>
				<phrase>solving methods, comparing</phrase>
			</indexes>
		</region>
		<region region-id="1412" left="6" top="1500.75" width="336.75" height="114" align-x="6" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#008080">• </c>
					<c val="#008080">
						<sp count="2"/>
					</c>If the equation you want to solve depends on variables defined above it in the worksheet, and you plan to update the values of the variables in the future, then you should use the symbolic equal sign with the <b>keyword</b>
					<inlineAttr font-weight="normal"> "solve</inlineAttr>." </p>
				<p style="List_custom" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit"/>
				<p style="List_custom" margin-left="14.25" margin-right="inherit" text-indent="-14.25" text-align="inherit" list-style-type="none" tabs="inherit">
					<c val="#008080">•</c>
					<c val="#008080">
						<sp/>
					</c>
					<sp count="2"/>If you do not want to display the symbolic equal sign or the keyword "solve," and you do not plan to update the values of variables in the worksheet, you can use <b>Variable &gt; Solve </b>from the <b>Symbolics</b> menu.</p>
			</text>
		</region>
		<region region-id="1434" left="0" top="1614" width="6000" height="6" align-x="0" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="1460" left="6" top="1628.25" width="50.25" height="12" align-x="6" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
			</text>
		</region>
		<region region-id="1461" left="6" top="1650.75" width="348.75" height="28.5" align-x="16.5" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Suppose you want to solve the following quadratic equation, with coefficients a, b, and c, for the variable z.</p>
			</text>
		</region>
		<region region-id="1463" left="18" top="1687.5" width="87.75" height="21" align-x="90.75" align-y="1704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve">a</ml:id>
								<ml:apply>
									<ml:pow/>
									<ml:id xml:space="preserve">z</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve">b</ml:id>
								<ml:id xml:space="preserve">z</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:id xml:space="preserve">c</ml:id>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="13"/>
		</region>
		<region region-id="1470" left="6" top="1716.75" width="351" height="28.5" align-x="9.75" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Since Mathcad assumes that a, b, c, and z are all variables, you must specify the variable you want to solve for as follows: </p>
			</text>
		</region>
		<region region-id="1545" left="6" top="1758.75" width="341.25" height="42.75" align-x="6" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>Place the cursor after the c in the equation above and click <b>solve</b> on the <b>Symbolic</b> toolbar to insert the keyword "solve" followed by a placeholder.</p>
			</text>
		</region>
		<region region-id="1630" left="18" top="1812" width="128.25" height="21.75" align-x="18" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<picture>
				<png item-idref="14" display-width="126.75" display-height="20.25"/>
			</picture>
			<rendering item-idref="15"/>
		</region>
		<region region-id="1478" left="6" top="1842.75" width="220.5" height="14.25" align-x="6" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<c val="#008080">
						<sp count="2"/>
					</c>
					<c val="#000">Type z in the placeholder and press </c>
					<b>
						<c val="#000">[Enter]</c>
					</b>
					<c val="#000">.</c>
					<c val="#008080">
						<sp count="2"/>
					</c>
				</p>
			</text>
		</region>
		<region region-id="1489" left="18" top="1872.75" width="234.75" height="123.75" align-x="133.5" align-y="1938" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve">a</ml:id>
								<ml:apply>
									<ml:pow/>
									<ml:id xml:space="preserve">z</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve">b</ml:id>
								<ml:id xml:space="preserve">z</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:id xml:space="preserve">c</ml:id>
					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:sequence>
					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:neg/>
								<ml:apply>
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									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:div/>
											<ml:id xml:space="preserve">b</ml:id>
											<ml:real>2</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:div/>
											<ml:apply>
												<ml:sqrt/>
												<ml:apply>
													<ml:minus/>
													<ml:apply>
														<ml:pow/>
														<ml:id xml:space="preserve">b</ml:id>
														<ml:real>2</ml:real>
													</ml:apply>
													<ml:apply>
														<ml:mult/>
														<ml:apply>
															<ml:mult/>
															<ml:real>4</ml:real>
															<ml:id xml:space="preserve">a</ml:id>
														</ml:apply>
														<ml:id xml:space="preserve">c</ml:id>
													</ml:apply>
												</ml:apply>
											</ml:apply>
											<ml:real>2</ml:real>
										</ml:apply>
									</ml:apply>
									<ml:id xml:space="preserve">a</ml:id>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:div/>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:div/>
											<ml:id xml:space="preserve">b</ml:id>
											<ml:real>2</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:div/>
											<ml:apply>
												<ml:sqrt/>
												<ml:apply>
													<ml:minus/>
													<ml:apply>
														<ml:pow/>
														<ml:id xml:space="preserve">b</ml:id>
														<ml:real>2</ml:real>
													</ml:apply>
													<ml:apply>
														<ml:mult/>
														<ml:apply>
															<ml:mult/>
															<ml:real>4</ml:real>
															<ml:id xml:space="preserve">a</ml:id>
														</ml:apply>
														<ml:id xml:space="preserve">c</ml:id>
													</ml:apply>
												</ml:apply>
											</ml:apply>
											<ml:real>2</ml:real>
										</ml:apply>
									</ml:apply>
									<ml:id xml:space="preserve">a</ml:id>
								</ml:apply>
							</ml:apply>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="16"/>
		</region>
		<region region-id="1491" left="6" top="2004.75" width="341.25" height="42.75" align-x="14.25" align-y="2016" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad returns the solutions symbolically using the quadratic formula. Now, suppose you want to solve the same equation for the following<sp count="2"/>values of the coefficients a, b, and c:</p>
			</text>
		</region>
		<region region-id="1449" left="42" top="2064.75" width="30" height="15.75" align-x="55.5" align-y="2076" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">a</ml:id>
					<ml:real>2</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="17"/>
		</region>
		<region region-id="1450" left="114" top="2064.75" width="30.75" height="15.75" align-x="128.25" align-y="2076" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">b</ml:id>
					<ml:real>3</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="18"/>
		</region>
		<region region-id="1451" left="186" top="2064.75" width="30" height="15.75" align-x="199.5" align-y="2076" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">c</ml:id>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="19"/>
		</region>
		<region region-id="1494" left="6" top="2088.75" width="331.5" height="71.25" align-x="17.25" align-y="2100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that defining these values for a, b, and c does not affect the preceding solution, because the variables are defined below the solution in the worksheet. However, if you retype the left side of the preceding solution below the definition of a, b, and c, Mathcad solves the equation for those values:</p>
			</text>
		</region>
		<region region-id="1447" left="36" top="2171.25" width="180" height="54.75" align-x="173.25" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:real>0</ml:real>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:plus/>
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									<ml:id xml:space="preserve">a</ml:id>
									<ml:apply>
										<ml:pow/>
										<ml:id xml:space="preserve">z</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:mult/>
									<ml:id xml:space="preserve">b</ml:id>
									<ml:id xml:space="preserve">z</ml:id>
								</ml:apply>
							</ml:apply>
							<ml:id xml:space="preserve">c</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:sequence>
					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:div/>
									<ml:real>1</ml:real>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:real>-1</ml:real>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="20"/>
		</region>
		<region region-id="1497" left="0" top="2232" width="6000" height="6" align-x="0" align-y="2232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="1496" left="6" top="2238.75" width="341.25" height="28.5" align-x="17.25" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you later change the value of a, b, or c anywhere above this expression in the worksheet, Mathcad updates the solution accordingly.</p>
			</text>
		</region>
		<region region-id="1407" left="6" top="2288.25" width="248.25" height="12" align-x="21.75" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving Equations Numerically</p>
			</text>
			<indexes>
				<phrase>solve for variable</phrase>
			</indexes>
		</region>
		<region region-id="1410" left="6" top="2304.75" width="340.5" height="28.5" align-x="9.75" align-y="2316" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you want a numerical solution to an equation, or if the equation has no symbolic solution, you can solve it using the <b>root</b> function.</p>
			</text>
		</region>
		<region region-id="308" left="6" top="2348.25" width="301.5" height="12" align-x="6" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="using root function">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using the root Function</p>
			</text>
			<indexes>
				<phrase>root function</phrase>
				<phrase>solving for roots</phrase>
			</indexes>
		</region>
		<region region-id="309" left="6" top="2364.75" width="341.25" height="14.25" align-x="6" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To solve an equation using the <b>root</b> function:</p>
			</text>
		</region>
		<region region-id="835" left="6" top="2394.75" width="341.25" height="28.5" align-x="6" align-y="2406" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>Define a guess value for the unknown variable. You can use a <inlineAttr font-weight="normal">graph</inlineAttr> to estimate the values of the solutions.</p>
			</text>
		</region>
		<region region-id="836" left="6" top="2430.75" width="341.25" height="28.5" align-x="6" align-y="2442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<sp count="2"/>Type the <b>root</b> function where the first argument is an expression equal to zero and the second argument is the unknown variable.</p>
			</text>
		</region>
		<region region-id="1250" left="6" top="2480.25" width="50.25" height="12" align-x="6" align-y="2490" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
			</text>
		</region>
		<region region-id="312" left="6" top="2493.75" width="163.5" height="34.5" align-x="6" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solve the equation <region region-id="702" left="92.25" top="2494.5" width="71.25" height="33.75" align-x="126.75" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:equal/>
								<ml:apply>
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve">x</ml:id>
								</ml:apply>
								<ml:apply>
									<ml:div/>
									<ml:apply>
										<ml:plus/>
										<ml:id xml:space="preserve">x</ml:id>
										<ml:real>1</ml:real>
									</ml:apply>
									<ml:real>10</ml:real>
								</ml:apply>
							</ml:apply>
						</math>
						<rendering item-idref="21"/>
					</region>. </p>
			</text>
		</region>
		<region region-id="704" left="6" top="2532.75" width="341.25" height="28.5" align-x="6" align-y="2544" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">It is a good idea to first plot the two functions to see approximately the number of times they intersect:</p>
			</text>
		</region>
		<region region-id="706" left="6" top="2574" width="236.25" height="202.5" align-x="6" align-y="2574" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<plot disable-calc="false" item-idref="22"/>
			<rendering item-idref="23"/>
		</region>
		<region region-id="1265" left="258" top="2586.75" width="154.5" height="14.25" align-x="264" align-y="2598" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>Tip</b>:<c val="#000"> To plot these functions, type: </c>
				</p>
			</text>
		</region>
		<region region-id="1268" left="258" top="2612.25" width="168" height="12" align-x="263.25" align-y="2622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<f family="Verdana" charset="0" size="10">
						<b>sin(x),x+1[space]/10@[Enter]</b>
					</f>
				</p>
			</text>
		</region>
		<region region-id="1362" left="258" top="2634.75" width="175.5" height="42.75" align-x="273.75" align-y="2646" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Double-click the graph to add gridlines if you want to make it easier to see where the roots are. </p>
			</text>
		</region>
		<region region-id="707" left="6" top="2790.75" width="341.25" height="42.75" align-x="6" align-y="2802" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">From the plot it looks like there are <i>seven</i> places where the two expressions are equal - the seven places where the blue line crosses the red sine curve. Use <b>root</b> to find these intersections.</p>
			</text>
		</region>
		<region region-id="1547" left="0" top="2838" width="6000" height="6" align-x="0" align-y="2838" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="856" left="6" top="2850.75" width="341.25" height="28.5" align-x="6" align-y="2862" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <b>root</b> function looks for places where an expression is equal to zero. So, first rearrange the equation into the form "something = 0."</p>
			</text>
		</region>
		<region region-id="1115" left="18" top="2890.5" width="90" height="33.75" align-x="93" align-y="2910" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:id xml:space="preserve">sin</ml:id>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
						<ml:apply>
							<ml:div/>
							<ml:apply>
								<ml:plus/>
								<ml:id xml:space="preserve">x</ml:id>
								<ml:real>1</ml:real>
							</ml:apply>
							<ml:real>10</ml:real>
						</ml:apply>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="24"/>
		</region>
		<region region-id="723" left="6" top="2934.75" width="341.25" height="73.5" align-x="6" align-y="2946" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <b>root </b>function finds only a single solution. Since there are actually seven solutions, you must indicate which one you are looking for. Do this by defining a guess value for the solution. Even if there's only one solution, <b>root</b> requires a guess value to get it started. First try to find the largest solution. From the graph it looks like it's around <region region-id="731" left="183" top="2992.5" width="27.75" height="15.75" align-x="195.75" align-y="3003.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:equal style="equal"/>
								<ml:id xml:space="preserve">x</ml:id>
								<ml:real>9</ml:real>
							</ml:apply>
						</math>
						<rendering item-idref="25"/>
					</region>, so use this guess:</p>
			</text>
		</region>
		<region region-id="1116" left="18" top="3018.75" width="30" height="15.75" align-x="31.5" align-y="3030" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:real>9</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="26"/>
		</region>
		<region region-id="1117" left="18" top="3040.5" width="162" height="33.75" align-x="137.25" align-y="3060" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">root</ml:id>
						<ml:sequence>
							<ml:apply>
								<ml:minus/>
								<ml:apply>
									<ml:id xml:space="preserve">sin</ml:id>
									<ml:id xml:space="preserve">x</ml:id>
								</ml:apply>
								<ml:apply>
									<ml:div/>
									<ml:apply>
										<ml:plus/>
										<ml:id xml:space="preserve">x</ml:id>
										<ml:real>1</ml:real>
									</ml:apply>
									<ml:real>10</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:sequence>
					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:real>8.2450444973817341</ml:real>
					</result>
				</ml:eval>
			</math>
			<rendering item-idref="27"/>
		</region>
		<region region-id="1107" left="6" top="3090.75" width="368.25" height="14.25" align-x="14.25" align-y="3102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Use a different guess value for x to try to find the smallest solution.</p>
			</text>
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				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:real>-9</ml:real>
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						<ml:real>-8.5665901023095952</ml:real>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving Systems of Equations</p>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving Two Equations by Setting Them Equal to One Another</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To solve two equations by setting them equal to each other:</p>
			</text>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>Type the two equations that make up the system using the Boolean equal sign. Type <b>[Ctrl] [=]</b> or click the Equal To button on the <b>Boolean</b> toolbar.</p>
			</text>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<sp count="2"/>Solve each equation using the symbolic equal sign with the keyword "solve."</p>
			</text>
		</region>
		<region region-id="866" left="6" top="3360.75" width="341.25" height="14.25" align-x="6" align-y="3372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">3.</c>
					</b>
					<sp count="2"/>Set the two solutions equal to one another using the Boolean equal sign.</p>
			</text>
		</region>
		<region region-id="869" left="6" top="3390.75" width="341.25" height="28.5" align-x="6" align-y="3402" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">4.</c>
					</b>
					<sp count="2"/>Solve for the remaining variable by clicking on any instance of it in the equation and choosing <b>Variable &gt; Solve</b> from the <b>Symbolics</b> menu.</p>
			</text>
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			<pageBreak/>
		</region>
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			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solve the following system of two equations and two unknowns:</p>
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			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:pow/>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:real>2</ml:real>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:real>2</ml:real>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>6</ml:real>
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			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:mult/>
							<ml:real>2</ml:real>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">y</ml:id>
					</ml:apply>
					<ml:real>3</ml:real>
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			</math>
			<rendering item-idref="31"/>
		</region>
		<region region-id="884" left="6" top="3528.75" width="341.25" height="42.75" align-x="15.75" align-y="3540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">First, solve each equation for the variable <inlineAttr font-style="normal">y</inlineAttr> by clicking at the end of the equation, then click <b>solve</b> on the <b>Symbolics</b> toolbar and type the variable y in the placeholder that appears after "solve."</p>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:id xml:space="preserve">x</ml:id>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal">Typing </inlineAttr>
					</f>
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						<math optimize="false" disable-calc="false">
							<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:id xml:space="preserve">x</ml:id>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:define>
						</math>
						<rendering item-idref="33"/>
					</region>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal"> and </inlineAttr>
					</f>
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						<math optimize="false" disable-calc="false">
							<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:id xml:space="preserve">y</ml:id>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:define>
						</math>
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					</region>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal"> prevents the Symbolic Processor from using the numeric values of x and y defined earlier in this worksheet. It does not change the numeric values of these variables.</inlineAttr>
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				</p>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
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								<ml:id xml:space="preserve">x</ml:id>
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							<ml:apply>
								<ml:mult/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
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						<ml:real>6</ml:real>
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					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:sequence>
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					<ml:symResult>
						<ml:apply>
							<ml:minus/>
							<ml:real>3</ml:real>
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:pow/>
									<ml:id xml:space="preserve">x</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="36"/>
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		<region region-id="1126" left="18" top="3720.75" width="150.75" height="15.75" align-x="124.5" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
						<ml:real>3</ml:real>
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					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
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					<ml:symResult>
						<ml:apply>
							<ml:minus/>
							<ml:real>3</ml:real>
							<ml:apply>
								<ml:mult/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
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		<region region-id="1517" left="6" top="3762.75" width="343.5" height="28.5" align-x="15" align-y="3774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To avoid retyping these solutions, you can copy and paste the two results setting them equal to each other with a <b>[Ctrl] [=]</b>. </p>
			</text>
		</region>
		<region region-id="1584" left="30" top="3802.5" width="92.25" height="33.75" align-x="79.5" align-y="3822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:real>3</ml:real>
						<ml:apply>
							<ml:mult/>
							<ml:apply>
								<ml:div/>
								<ml:real>1</ml:real>
								<ml:real>2</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:pow/>
								<ml:id xml:space="preserve">x</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:apply>
					<ml:apply>
						<ml:minus/>
						<ml:real>3</ml:real>
						<ml:apply>
							<ml:mult/>
							<ml:real>2</ml:real>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
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		<region region-id="1586" left="6" top="3846.75" width="335.25" height="28.5" align-x="17.25" align-y="3858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click at the end of the equation and solve the equation by clicking on the keyword "solve" from the <b>Symbolic</b> toolbar.</p>
			</text>
		</region>
		<region region-id="1212" left="6" top="3888.75" width="341.25" height="28.5" align-x="6" align-y="3900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Finally, set the two solutions equal to each other using the Boolean equal sign and solve this equation using the keyword "solve": </p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:minus/>
							<ml:real>3</ml:real>
							<ml:apply>
								<ml:mult/>
								<ml:apply>
									<ml:div/>
									<ml:real>1</ml:real>
									<ml:real>2</ml:real>
								</ml:apply>
								<ml:apply>
									<ml:pow/>
									<ml:id xml:space="preserve">x</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:minus/>
							<ml:real>3</ml:real>
							<ml:apply>
								<ml:mult/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:apply>
					<ml:command>
						<ml:id xml:space="preserve">solve</ml:id>
					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:real>4</ml:real>
							<ml:real>0</ml:real>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="39"/>
		</region>
		<region region-id="1592" left="6" top="3972.75" width="343.5" height="42.75" align-x="14.25" align-y="3984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The empty placeholder that may appear is for modifiers. If it is not needed, then just backspace over it. You can read about keyword modifiers in Online Help.</p>
			</text>
		</region>
		<region region-id="1565" left="0" top="4020" width="6000" height="6" align-x="0" align-y="4020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="1526" left="6" top="4020.75" width="341.25" height="42.75" align-x="6" align-y="4032" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To graph the original two equations and their solutions: </p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>Define a range variable x over which to plot the two expressions.</p>
			</text>
		</region>
		<region region-id="1566" left="18" top="4068.75" width="92.25" height="15.75" align-x="31.5" align-y="4080" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:range>
						<ml:sequence>
							<ml:real>-10</ml:real>
							<ml:real>-9.9</ml:real>
						</ml:sequence>
						<ml:real>10</ml:real>
					</ml:range>
				</ml:define>
			</math>
			<rendering item-idref="40"/>
		</region>
		<region region-id="1528" left="6" top="4092.75" width="408.75" height="14.25" align-x="18" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<sp count="2"/>Type <b>[@]</b> to create a blank graph, then insert x into the x-axis placeholder. </p>
			</text>
		</region>
		<region region-id="1567" left="6" top="4116.75" width="341.25" height="28.5" align-x="6" align-y="4128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit">
					<b>
						<c val="#008080">3.</c>
					</b>
					<sp count="2"/>In the y-axis placeholder, type the first expression, followed by a comma, then the second expression. </p>
			</text>
		</region>
		<region region-id="1570" left="6" top="4158.75" width="199.5" height="14.25" align-x="9.75" align-y="4170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">4.</c>
					</b>
					<sp count="2"/>Double-click the graph to add gridlines.<sp count="2"/>
				</p>
			</text>
		</region>
		<region region-id="1573" left="6" top="4188.75" width="341.25" height="28.5" align-x="18.75" align-y="4200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The two points where the graphs intersect are the two solutions to the system of equations.</p>
			</text>
		</region>
		<region region-id="1568" left="12" top="4230" width="322.5" height="214.5" align-x="12" align-y="4230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<plot disable-calc="false" item-idref="41"/>
			<rendering item-idref="42"/>
		</region>
		<region region-id="1574" left="0" top="4458" width="6000" height="6" align-x="0" align-y="4458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="772" left="6" top="4466.25" width="310.5" height="12" align-x="6" align-y="4476" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="solve block">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving a System of Equations Using a Solve Block</p>
			</text>
			<indexes>
				<phrase>solve blocks</phrase>
				<phrase>Find function</phrase>
			</indexes>
		</region>
		<region region-id="773" left="6" top="4488.75" width="341.25" height="42.75" align-x="6" align-y="4500" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A <b>Solve Block</b> can be used to solve any solvable system of equations with any number of equations and unknowns. Here are the steps for setting up a Solve Block:</p>
			</text>
		</region>
		<region region-id="1548" left="6" top="4536.75" width="341.25" height="42.75" align-x="6" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">1.</c>
					</b>
					<sp count="2"/>In your Worksheet window, define guess values for the unknown variables using a definition equal sign. To get the definition equal sign, type<b> :</b> (colon) or click the Definition button on the <b>Calculator</b> toolbar.</p>
			</text>
		</region>
		<region region-id="1014" left="6" top="4596.75" width="341.25" height="14.25" align-x="6" align-y="4608" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">2.</c>
					</b>
					<sp count="2"/>Type <b>given</b> (as in "given these equations") as a separate math region.</p>
			</text>
			<indexes>
				<phrase>given</phrase>
			</indexes>
		</region>
		<region region-id="1015" left="6" top="4620.75" width="368.25" height="14.25" align-x="6" align-y="4632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">3.</c>
					</b>
					<sp count="2"/>Enter the equations using<b> [Ctrl] [=]</b>.</p>
			</text>
		</region>
		<region region-id="1016" left="6" top="4644.75" width="345.75" height="28.5" align-x="6" align-y="4656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">4.</c>
					</b>
					<sp count="2"/>Type <b>F</b>
					<b>ind</b>
					<inlineAttr font-weight="normal">,</inlineAttr> then enter all the unknown variables in parentheses following <b>F</b>
					<b>ind</b>. Type <b>[=]</b> to get the results. For example, type <b>F</b>
					<b>ind(x,y,z)=</b>.</p>
			</text>
		</region>
		<region region-id="1017" left="0" top="4680" width="6000" height="6" align-x="0" align-y="4680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="1261" left="6" top="4700.25" width="50.25" height="12" align-x="6" align-y="4710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
			</text>
		</region>
		<region region-id="1213" left="6" top="4716.75" width="314.25" height="14.25" align-x="18" align-y="4728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solve the following system of three equations and three unknowns:</p>
			</text>
		</region>
		<region region-id="1227" left="18" top="4752.75" width="89.25" height="15.75" align-x="92.25" align-y="4764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>3</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:real>4</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="auto-select"/>
							<ml:real>4</ml:real>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>2</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="43"/>
		</region>
		<region region-id="1228" left="126" top="4752.75" width="81" height="15.75" align-x="192" align-y="4764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>-7</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">z</ml:id>
					</ml:apply>
					<ml:real>5</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="44"/>
		</region>
		<region region-id="1229" left="222" top="4752.75" width="75.75" height="15.75" align-x="282.75" align-y="4764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>6</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>3</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:id xml:space="preserve">z</ml:id>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="45"/>
		</region>
		<region region-id="1131" left="18" top="4782.75" width="30" height="15.75" align-x="31.5" align-y="4794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="46"/>
		</region>
		<region region-id="1132" left="60" top="4782.75" width="36.75" height="15.75" align-x="73.5" align-y="4794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
					<ml:real>-1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="47"/>
		</region>
		<region region-id="1133" left="108" top="4782.75" width="29.25" height="15.75" align-x="120.75" align-y="4794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">z</ml:id>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="48"/>
		</region>
		<region region-id="1134" left="162" top="4782.75" width="105.75" height="14.25" align-x="162" align-y="4794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<inlineAttr font-weight="normal">Guess values</inlineAttr>
				</p>
			</text>
		</region>
		<region region-id="1135" left="18" top="4818.75" width="27" height="15.75" align-x="30.75" align-y="4830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id>
			</math>
			<rendering item-idref="49"/>
		</region>
		<region region-id="1136" left="162" top="4818.75" width="201.75" height="14.25" align-x="162" align-y="4830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<inlineAttr font-weight="normal">The word </inlineAttr>
					<b>G</b>
					<b>iven</b>
					<inlineAttr font-weight="normal"> in a separate math region</inlineAttr>
				</p>
			</text>
		</region>
		<region region-id="1182" left="42" top="4842.75" width="89.25" height="15.75" align-x="116.25" align-y="4854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>3</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:real>4</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="auto-select"/>
							<ml:real>4</ml:real>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>2</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="50"/>
		</region>
		<region region-id="1138" left="162" top="4860.75" width="109.5" height="14.25" align-x="162" align-y="4872" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The equations</p>
			</text>
		</region>
		<region region-id="1183" left="42" top="4866.75" width="81" height="15.75" align-x="108" align-y="4878" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>-7</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">z</ml:id>
					</ml:apply>
					<ml:real>5</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="51"/>
		</region>
		<region region-id="1184" left="42" top="4890.75" width="75.75" height="15.75" align-x="102.75" align-y="4902" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>6</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>3</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:id xml:space="preserve">z</ml:id>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="52"/>
		</region>
		<region region-id="1141" left="18" top="4917.75" width="130.5" height="57.75" align-x="82.5" align-y="4950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/>
						<ml:sequence>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:sequence>
					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:matrix rows="3" cols="1">
							<ml:real>-0.52173913043478259</ml:real>
							<ml:real>1.1195652173913042</ml:real>
							<ml:real>0.2282608695652173</ml:real>
						</ml:matrix>
					</result>
				</ml:eval>
			</math>
			<rendering item-idref="53"/>
		</region>
		<region region-id="1142" left="162" top="4938.75" width="135.75" height="14.25" align-x="162" align-y="4950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>Find</b> statement </p>
			</text>
		</region>
		<region region-id="1532" left="0" top="4980" width="6000" height="6" align-x="0" align-y="4980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="1022" left="6" top="4986" width="351" height="72.75" align-x="15" align-y="4998" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can use the symbolic equal sign (<f family="Symbol" charset="2">®</f>), by typing <b>[Ctrl] [.]</b>
					<inlineAttr font-weight="normal">,</inlineAttr> to display a result in a Solve Block. When using the symbolic equal sign, you do not need to supply guess values for the variables. You get an exact numeric result rather than a decimal result with your chosen degree of precision. Here's the same Solve Block, using <f family="Symbol" charset="2">® </f>
					<f family="Times New Roman" charset="0">for the results</f>.</p>
			</text>
		</region>
		<region region-id="1349" left="18" top="5082.75" width="30" height="15.75" align-x="31.5" align-y="5094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
					<ml:real>-1</ml:real>
				</ml:define>
			</math>
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		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">z</ml:id>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="56"/>
		</region>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<inlineAttr font-weight="normal">Guess values</inlineAttr>
				</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id>
			</math>
			<rendering item-idref="57"/>
		</region>
		<region region-id="1151" left="162" top="5106.75" width="162" height="14.25" align-x="162" align-y="5118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The word <b>Given</b> as a math region</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>3</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:real>4</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="auto-select"/>
							<ml:real>4</ml:real>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>2</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="58"/>
		</region>
		<region region-id="1153" left="162" top="5148.75" width="70.5" height="14.25" align-x="162" align-y="5160" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The equations</p>
			</text>
		</region>
		<region region-id="1186" left="30" top="5154.75" width="81" height="15.75" align-x="96" align-y="5166" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:real>-7</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">z</ml:id>
					</ml:apply>
					<ml:real>5</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="59"/>
		</region>
		<region region-id="1187" left="30" top="5178.75" width="75.75" height="15.75" align-x="90.75" align-y="5190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>6</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>3</ml:real>
								<ml:id xml:space="preserve">y</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:id xml:space="preserve">z</ml:id>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="60"/>
		</region>
		<region region-id="1156" left="18" top="5208.75" width="114" height="111.75" align-x="83.25" align-y="5268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/>
						<ml:sequence>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:sequence>
					</ml:apply>
					<ml:symResult>
						<ml:matrix rows="3" cols="1">
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:div/>
									<ml:real>12</ml:real>
									<ml:real>23</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>103</ml:real>
								<ml:real>92</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>21</ml:real>
								<ml:real>92</ml:real>
							</ml:apply>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="61"/>
		</region>
		<region region-id="1576" left="162" top="5256.75" width="75.75" height="14.25" align-x="162" align-y="5268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>Find</b> statement </p>
			</text>
		</region>
		<region region-id="1363" left="6" top="5340.75" width="341.25" height="28.5" align-x="18" align-y="5352" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">It is useful to use the symbolic equal sign if you want a symbolic solution rather than a numeric one. Here's an example:</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">x</ml:id>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="62"/>
		</region>
		<region region-id="1358" left="60" top="5388.75" width="36.75" height="15.75" align-x="73.5" align-y="5400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
					<ml:real>-1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="63"/>
		</region>
		<region region-id="1360" left="150" top="5388.75" width="105.75" height="14.25" align-x="150" align-y="5400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<inlineAttr font-weight="normal">Guess values</inlineAttr>
				</p>
			</text>
		</region>
		<region region-id="1143" left="18" top="5412.75" width="27" height="15.75" align-x="30.75" align-y="5424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id>
			</math>
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		</region>
		<region region-id="1144" left="150" top="5412.75" width="165" height="14.25" align-x="150" align-y="5424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The word <b>Given</b> as a math region</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">a</ml:id>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">b</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>2</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="65"/>
		</region>
		<region region-id="1233" left="150" top="5454.75" width="169.5" height="14.25" align-x="150" align-y="5466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The equations using <b>
						<c val="#008080">[Ctrl] [=]</c>
					</b>
				</p>
			</text>
		</region>
		<region region-id="1361" left="30" top="5460.75" width="65.25" height="15.75" align-x="80.25" align-y="5472" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">c</ml:id>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">d</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>5</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="66"/>
		</region>
		<region region-id="1577" left="156" top="5520.75" width="161.25" height="28.5" align-x="156" align-y="5532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type <b>F</b>
					<b>
						<c val="#008080">ind(x,y)[Ctrl] [.]</c>
					</b> to display a symbolic result</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/>
						<ml:sequence>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:sequence>
					</ml:apply>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:mult/>
										<ml:real>2</ml:real>
										<ml:id xml:space="preserve">d</ml:id>
									</ml:apply>
									<ml:real>15</ml:real>
								</ml:apply>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:mult/>
										<ml:real>2</ml:real>
										<ml:id xml:space="preserve">d</ml:id>
									</ml:apply>
									<ml:real>3</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>8</ml:real>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:mult/>
										<ml:real>2</ml:real>
										<ml:id xml:space="preserve">d</ml:id>
									</ml:apply>
									<ml:real>3</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="67"/>
		</region>
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			<picture>
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