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<h2 data-type="title">Evaluating Algebraic Expressions</h2>
<p>In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.</p>
<p>To <strong>evaluate</strong> an algebraic expression means to find the value of the expression when the is replaced by a given number. To evaluate an , we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.</p>
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<h2>Example</h2>
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<p>Evaluate \(x+7\) when</p>
<ol type="1">
<li>\(\phantom{\rule{0.2em}{0ex}}x=3\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}x=12\)</li>
</ol>
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<h3 data-type="title">Solution</h3>
<p>To evaluate, substitute \(3\) for \(x\) in the expression, and then simplify.</p>
<table data-label="" summary="The image shows the given expression x plus 7. Substitute 3 for x. The expression becomes 3 plus x which is 10.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/9580b0f9-3397-40f8-8a6a-04f39e9e25db"/></td>
</tr>
<tr>
<td>Substitute.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/04ae4e60-2b25-4e61-b57b-e28b8565da18"/></td>
</tr>
<tr>
<td>Add.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/58554c8b-ad10-4495-9a07-4622a9e8a80e"/></td>
</tr>
</tbody>
</table>
<p>When \(x=3,\) the expression \(x+7\) has a value of \(10.\)</p>
<p>To evaluate, substitute \(12\) for \(x\) in the expression, and then simplify.</p>
<table data-label="" summary="The image shows the given expression x plus 7, substitute 12 for x. The expression becomes 12 plus x which is 19.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/8bd54246-c811-4a47-8b3a-fb239ec7c7d6"/></td>
</tr>
<tr>
<td>Substitute.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/1edccf1e-ece4-4099-8e6e-2508c34a3d6a"/></td>
</tr>
<tr>
<td>Add.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/70947dac-2a65-49dd-b057-3a253fe768de"/></td>
</tr>
</tbody>
</table>
<p>When \(x=12,\) the expression \(x+7\) has a value of \(19.\)</p>
<p>Notice that we got different results for parts and even though we started with the same expression. This is because the values used for \(x\) were different. When we evaluate an expression, the value varies depending on the value used for the variable.</p>
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<h2>Example</h2>
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<p>Evaluate \(9x-2,\text{when}\phantom{\rule{0.2em}{0ex}}\)</p>
<ol type="1">
<li>\(\phantom{\rule{0.2em}{0ex}}x=5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}x=1\)</li>
</ol>
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<h3 data-type="title">Solution</h3>
<p>Remember \(ab\) means \(a\) times \(b,\) so \(9x\) means \(9\) times \(x.\)</p>
<p>To evaluate the expression when \(x=5,\) we substitute \(5\) for \(x,\) and then simplify.</p>
<table data-label="" summary="The image shows the given expression nine x minus 2. Substitute 5 for x. The expression becomes 9 times 5 minus 2. Multiply first. Nine times 5 is 45 and the expression is now 45 minus 2. Subtract to get 43.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/e23f6879-3274-4b0e-8158-00bc0a85370f"/></td>
</tr>
<tr>
<td><img alt="." src="https://data.ulearngo.com/assets/9bb88139-2bbf-459e-9cfa-90b8f8f236e9"/></td>
<td><img alt="." src="https://data.ulearngo.com/assets/30ff605a-882a-4cf4-9761-ca5059f98952"/></td>
</tr>
<tr>
<td>Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/efe32e9a-caa2-4ef9-9e46-4799b57c9db7"/></td>
</tr>
<tr>
<td>Subtract.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/76961477-4281-436f-903d-9ac93bed516d"/></td>
</tr>
</tbody>
</table>
<p>To evaluate the expression when \(x=1,\) we substitute \(1\) for \(x,\) and then simplify.</p>
<table data-label="" summary="The image shows the given expression nine x minus 2. Substitute 1 for x. The expression becomes 9 times 1 minus 2. Multiply first. Nine times 1 is 9 and the expression is now 9 minus 2. Subtract to get 7.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/d5ca6756-aaf9-4a38-85c8-f8d39f5af3fa"/></td>
</tr>
<tr>
<td><img alt="." src="https://data.ulearngo.com/assets/7be19eb3-bd39-4301-ba47-f4cc15ed0cc9"/></td>
<td><img alt="." src="https://data.ulearngo.com/assets/6f91c4ce-4185-47d9-ac48-68c921502118"/></td>
</tr>
<tr>
<td>Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/4a7a5dc1-16ec-4b4c-a357-246c07ac5bf8"/></td>
</tr>
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<td>Subtract.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/10566d54-d1b8-49da-ae30-fbcfb1a6093b"/></td>
</tr>
</tbody>
</table>
<p>Notice that in part that we wrote \(9\cdot 5\) and in part we wrote \(9\left(1\right).\) Both the dot and the parentheses tell us to multiply.</p>
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<h2>Example</h2>
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<p>Evaluate \({x}^{2}\) when \(x=10.\) </p>
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<h3 data-type="title">Solution</h3>
<p>We substitute \(10\) for \(x,\) and then simplify the expression.</p>
<table data-label="" summary="The image shows the given expression x squared. Substitute 10 for x. The expression becomes 10 squared. By the definition of exponents, 10 squared is 10 times 10. Multiply to get 100.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/d05f214f-4d08-4b5f-a8fa-ed782355fb7f"/></td>
</tr>
<tr>
<td><img alt="." src="https://data.ulearngo.com/assets/3c9e1142-dda4-4d40-8807-b1c40b0a58d4"/></td>
<td><img alt="." src="https://data.ulearngo.com/assets/ed8a1cac-f54b-443a-a125-8f12fcc90143"/></td>
</tr>
<tr>
<td>Use the definition of exponent.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/814286dd-9078-4a32-8a82-33171dba1631"/></td>
</tr>
<tr>
<td>Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/0a03eadd-8eb2-4f8a-b80e-45986f0b792f"/></td>
</tr>
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<p>When \(x=10,\) the expression \({x}^{2}\) has a value of \(100.\)</p>
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<h2>Example</h2>
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<p>\(\text{Evaluate}\phantom{\rule{0.2em}{0ex}}{2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=5.\)</p>
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<h3 data-type="title">Solution</h3>
<p>In this expression, the variable is an exponent.</p>
<table data-label="" summary="The image shows the given expression 2 to the power of x. Substitute 5 for x. The expression becomes 2 to the fifth power. By the definition of exponents, 2 to the fifth power is 2 times 2 times 2 times 2 times 2, or 5 factors of 2. Multiply from left to right to get 32.">
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<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/56231301-4aad-4398-a3b0-e5efab0c8785"/></td>
</tr>
<tr>
<td><img alt="." src="https://data.ulearngo.com/assets/4bc7bdb7-0833-4f60-a869-546a93abc903"/></td>
<td><img alt="." src="https://data.ulearngo.com/assets/66b42f17-4388-4e86-822a-e3964494b4f6"/></td>
</tr>
<tr>
<td>Use the definition of exponent.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/3a270efc-4539-4d4d-8ecd-2a65c125bedd"/></td>
</tr>
<tr>
<td>Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/215dd48c-9410-4f9e-b798-dab55610ab3c"/></td>
</tr>
</tbody>
</table>
<p>When \(x=5,\) the expression \({2}^{x}\) has a value of \(32.\)</p>
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<h2>Example</h2>
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<div data-type="example">
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<p>\(\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2.\)</p>
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<h3 data-type="title">Solution</h3>
<p>This expression contains two variables, so we must make two substitutions.</p>
<table data-label="" summary="The image shows the given expression three x plus four y minus 6. Substitute 10 for x and 2 for y. The expression becomes 3 times 10 plus 4 times 2 minus 6. Perform multiplication from left to right. Three times 10 is 30 and 4 times 2 is 8. The expression becomes 30 plus 8 minus 6. Add and subtract from left to right. Thirty plus 8 is 38. Thirty-eight minus 6 is 32.">
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<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/151804fd-19fb-4001-baf0-6c6432c3ea7f"/></td>
</tr>
<tr>
<td><img alt="." src="https://data.ulearngo.com/assets/378a9fbb-f64a-4811-87b0-ae45f6c8a8bd"/></td>
<td><img alt="." src="https://data.ulearngo.com/assets/06a11775-4505-456a-8abd-58a579288577"/></td>
</tr>
<tr>
<td>Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/bd30d949-af71-4c65-9104-bc0cd6c68472"/></td>
</tr>
<tr>
<td>Add and subtract left to right.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/f8e1d9ef-c59a-4907-b651-b64db481e65f"/></td>
</tr>
</tbody>
</table>
<p>When \(x=10\) and \(y=2,\) the expression \(3x+4y-6\) has a value of \(32.\)</p>
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<h2>Example</h2>
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<p>\(\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}+3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4.\)</p>
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<h3 data-type="title">Solution</h3>
<p>We need to be careful when an expression has a variable with an exponent. In this expression, \(2{x}^{2}\) means \(2\cdot x\cdot x\) and is different from the expression \({\left(2x\right)}^{2},\) which means \(2x\cdot 2x.\)</p>
<table data-label="" summary="The image shows the given expression two x squared plus three x plus 8. Substitute 4 for  each x. The expression becomes 2 times 4 squared plus 3 times 4 plus 8. Simplify exponents first. Four squared is 16 so the expression becomes 2 times 16 plus 3 times 4 plus 8. Next perform multiplication from left to right. Two times 16 is 32 and 3 times 4 is 12. The expression becomes 32 plus 12 plus 8. Add from left to right. Thirty-two plus 12 is 44. Forty-four plus 8 is 52.">
<tbody>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/96d96f63-160b-439f-a3d4-490029fa72f0"/></td>
</tr>
<tr>
<td><img alt="." src="https://data.ulearngo.com/assets/3b0fb62a-78dd-4abc-950e-770ae9228812"/></td>
<td><img alt="." src="https://data.ulearngo.com/assets/5cdc7623-e332-4e51-92fb-b0887599462e"/></td>
</tr>
<tr>
<td>Simplify \({4}^{2}\).</td>
<td><img alt="." src="https://data.ulearngo.com/assets/ec40c17a-03fc-4e4b-a650-48bf476e263d"/></td>
</tr>
<tr>
<td>Multiply.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/9f56ae0c-3d92-41e7-a008-839fe9a65f48"/></td>
</tr>
<tr>
<td>Add.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/5f27ffd8-1e3c-49a9-bb95-1f0e072a73e3"/></td>
</tr>
</tbody>
</table>
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Evaluating Algebraic Expressions

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Learn to identify and simplify algebraic expressions, evaluate expressions, and translate word problems into algebraic equations in this comprehensive tutorial on the language of algebra.


Evaluating Algebraic Expressions

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