Equation is a value of a variable that maker a true statement when substitutee into the equation. For Ex. Substitute 3/4?

<p>We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problem-solving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.</p>
<h2>Solving Equations Using the Subtraction and Addition Properties of Equality</h2>
<p>We began our work solving equations in previous tutorials. It has been a while since we have seen an <strong>equation</strong>, so we will review some of the key concepts before we go any further.</p>
<p>We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the <strong>variable</strong> that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Definition: Solution of an Equation</h3>
<p>A <strong>solution of an equation</strong> is a value of a variable that makes a true statement when substituted into the equation.</p>
</div>
</div>
<p>In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>How to Determine whether a number is a solution to an equation.</h3>
<ol type="1">
<li>Substitute the number for the variable in the equation.</li>
<li>Simplify the expressions on both sides of the equation.</li>
<li>Determine whether the resulting equation is true.

<ul data-bullet-style="bullet" style="list-style-type: square;">
<li>If it is true, the number is a solution.</li>
<li>If it is not true, the number is not a solution.</li>
</ul>
</li>
</ol>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Determine whether \(y=\frac{3}{4}\) is a solution for \(4y+3=8y.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" style="width: 414px;" summary="The top line says 4y plus 3 equals 8y. Beside this is “Given.” The next line says 4 times a red three-fourths plus 3 followed by an equal sign with a question mark, then 8 times a red three-fourths. Beside this is “Substitute three-fourths for y.” Three-fourths is in red. The next line says 3 plus 3 equal sign with a question mark 6. Beside this is “Multiply.” The last line says 6 equals 6. Beside this is “Add. The solution checks.">
<tbody>
<tr>
<td style="width: 230px;"> </td>
<td style="width: 178px;"><img alt="." src="https://data.ulearngo.com/assets/e72b5b7c-a79b-4a2e-9afc-6e6495c96fc8"/></td>
</tr>
<tr>
<td style="width: 230px;"><img alt="." src="https://data.ulearngo.com/assets/5dcf38ec-f9c2-4373-8ed5-dd770a9e88d6"/></td>
<td style="width: 178px;"><img alt="." src="https://data.ulearngo.com/assets/2d662b2e-d3a3-46ba-b67b-ccfb0305c081"/></td>
</tr>
<tr>
<td style="width: 230px;">Multiply.</td>
<td style="width: 178px;"><img alt="." src="https://data.ulearngo.com/assets/435af1a8-2e91-41b5-9d4f-33a0c495528a"/></td>
</tr>
<tr>
<td style="width: 230px;">Add.</td>
<td style="width: 178px;"><img alt="." src="https://data.ulearngo.com/assets/4fc2d187-8b44-4215-bba3-86f3c3d0c496"/></td>
</tr>
</tbody>
</table>
<p>Since \(y=\frac{3}{4}\) results in a true equation, \(\frac{3}{4}\) is a solution to the equation \(4y+3=8y.\)</p>
</div>
</div>
</div>
</div>
</div>
<p>We introduced the Subtraction and Addition Properties of Equality earlier. We modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Definition: Subtraction and Addition Properties of Equality</h3>
<p><strong>Subtraction Property of Equality</strong></p>
<p>For all real numbers \(a,b,\) and \(c,\) if \(a=b,\) then \(a-c=b-c.\)</p>
<p><strong>Addition Property of Equality</strong></p>
<p>For all real numbers \(a,b,\) and \(c,\) if \(a=b,\) then \(a+c=b+c.\)</p>
</div>
</div>
<p>When you add or subtract the same quantity from both sides of an equation, you still have equality.</p>
<p>We introduced the <strong>Subtraction Property of Equality</strong> earlier by modeling equations with envelopes and counters. The figure below models the equation \(x+3=8.\)</p>
<p><img alt="An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters." src="https://data.ulearngo.com/assets/ef690983-2df1-4a8f-885b-c6f634241785"/></p>
<p>The goal is to isolate the variable on one side of the equation. So we ‘took away’ \(3\) from both sides of the equation and found the <strong>solution</strong> \(x=5.\)</p>
<p>Some people picture a balance scale, as in the figure below, when they solve equations.</p>
<p><img alt="Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is “1 mass on each side equals balanced.” The next scale has two weights on each side and is balanced. Beside it is “2 masses on each side equals balanced.” The bottom scale has one weight on the left and two on the right. The right side is lower than the left. Beside the image is “1 mass on one side and 2 masses on the other equals unbalanced.”" src="https://data.ulearngo.com/assets/cfd46977-dbca-4e1d-8dd1-2ed6d7208845"/></p>
<p>The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.</p>
<p>Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(x+11=-3.\)</p>
</div>
<div>
<h3>Solution</h3>
<p>To isolate \(x,\) we undo the addition of \(11\) by using the Subtraction Property of Equality.</p>
<table data-label="" style="width: 630px;" summary="The top line says x plus 11 equals negative 3. The next line says, “Subtract 11 from each side, to 'undo' the addition.” Beside this is x plus 11 minus a red 11 equals negative 3 minus a red 11. The next line says “simplify.” Beside this is x equals negative 14. The next line says “check.” Beside this is x plus 11 equals negative 3. The last line says “substitute” and shows x equals negative 14, then a red negative 14 plus 11 followed by an equal sign with a question mark and a negative 3, then negative 3 equals negative 3 followed by a check mark.">
<tbody>
<tr>
<td colspan="2" style="width: 417px;"> </td>
<td style="width: 205px;"><img alt="." src="https://data.ulearngo.com/assets/32970730-48fa-4497-8d3c-ecc2576d7b8b"/></td>
</tr>
<tr>
<td colspan="2" style="width: 417px;">Subtract 11 from each side to "undo" the addition.</td>
<td style="width: 205px;"><img alt="." src="https://data.ulearngo.com/assets/8ca994f9-5f1f-4d92-a502-3dcae2c8215c"/></td>
</tr>
<tr>
<td colspan="2" style="width: 417px;">Simplify.</td>
<td style="width: 205px;"><img alt="." src="https://data.ulearngo.com/assets/2379e2e6-493f-4805-8c1f-ec83b9ce4b6a"/></td>
</tr>
<tr>
<td style="width: 213px;">Check:</td>
<td style="width: 204px;"><img alt="." src="https://data.ulearngo.com/assets/afb75ce1-b296-4218-b340-3ed4ac169344"/></td>
<td style="width: 205px;"> </td>
</tr>
<tr>
<td style="width: 213px;">Substitute \(x=-14\).</td>
<td style="width: 204px;"><img alt="." src="https://data.ulearngo.com/assets/599940da-19a0-4b8c-bf12-4acbeb12e1c0"/></td>
<td style="width: 205px;"> </td>
</tr>
<tr>
<td style="width: 213px;"> </td>
<td style="width: 204px;"><img alt="." src="https://data.ulearngo.com/assets/232ca81c-9988-4992-8429-1f481eaea532"/></td>
<td style="width: 205px;"> </td>
</tr>
</tbody>
</table>
<p>Since \(x=-14\) makes \(x+11=-3\) a true statement, we know that it is a solution to the equation.</p>
</div>
</div>
</div>
</div>
</div>
<p>In the original equation in the previous example, \(11\) was added to the \(x\), so we subtracted \(11\) to ‘undo’ the addition. In the next example, we will need to ‘undo’ subtraction by using the <strong>Addition Property of Equality</strong>.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(m-4=-5.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" style="width: 706px;" summary="The first line says m minus 4 equals negative 5. The next line says, “Add 4 to each side, to 'undo' the subtraction.” Beside this is m minus 4 plus a red 4 equals negative 5 plus a red 4. The next line says “simplify” and shows m equals negative 1. The following line says to “check” and shows m minus 4 equals negative 5. Below this is “substitute” and m equals negative 1. Below that is a red negative 1 minus 4 followed by an equal sign with a question mark and negative 5. The last line shows negative 5 equals negative 5 followed by a check mark.">
<tbody>
<tr>
<td colspan="2" style="width: 354px;"> </td>
<td style="width: 334px;"><img alt="." src="https://data.ulearngo.com/assets/05bedf70-b18d-4dc9-98bc-fed05b2b3548"/></td>
</tr>
<tr>
<td colspan="2" style="width: 354px;">Add 4 to each side to "undo" the subtraction.</td>
<td style="width: 334px;"><img alt="." src="https://data.ulearngo.com/assets/914e41ff-9d6d-4b21-9d2a-277259713d9e"/></td>
</tr>
<tr>
<td colspan="2" style="width: 354px;">Simplify.</td>
<td style="width: 334px;"><img alt="." src="https://data.ulearngo.com/assets/0aa6881a-1b18-4f43-b2cc-662b169dd371"/></td>
</tr>
<tr>
<td style="width: 173px;">Check:</td>
<td style="width: 181px;"><img alt="." src="https://data.ulearngo.com/assets/45c68f64-832d-4146-b50f-cca74329963e"/></td>
<td style="width: 334px;"> </td>
</tr>
<tr>
<td style="width: 173px;">Substitute \(m=-1\).</td>
<td style="width: 181px;"><img alt="." src="https://data.ulearngo.com/assets/8181a241-c1c2-4026-8699-5d6af86912cb"/></td>
<td style="width: 334px;"> </td>
</tr>
<tr>
<td style="width: 173px;"> </td>
<td style="width: 181px;"><img alt="." src="https://data.ulearngo.com/assets/a16b9712-22c8-461f-a9de-4277c5b4f327"/></td>
<td style="width: 334px;"> </td>
</tr>
<tr>
<td style="width: 173px;"> </td>
<td style="width: 181px;"> </td>
<td style="width: 334px;">The solution to \(m-4=-5\) is \(m=-1\).</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>Now let’s review solving equations with fractions.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve: \(n-\frac{3}{8}=\frac{1}{2}.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" style="width: 569px;" summary="The first line says n minus 3 eighths equals one half. The next line says, “Use the Addition Property of Equality.” Beside this is n minus 3 over 8 plus a red 3 over 8 equals one half plus a red 3 over 8. The next line says “Find the LCD to add the fractions on the right.” Beside that is n minus 3 over 8 plus 3 over 8 equals 4 over 8 plus 3 over 8. The next line says “simplify” and shows n equals 7 over 8. Below that is “check” and n minus 3 eighths equals one half. The following line says to substitute n equals 7 over 8 and shows a red 7 over 8 minus 3 over 8 followed by an equal sign with a question mark and one half. The next line says “subtract” and shows 4 over 8, equal sign with a question mark, 1 half. The last line says “simplify” and says one half equals one half with a check mark.">
<tbody>
<tr>
<td colspan="2" style="width: 366px;"> </td>
<td style="width: 197px;"><img alt="." src="https://data.ulearngo.com/assets/e5a382ab-9e4c-4554-a735-8648e099c922"/></td>
</tr>
<tr>
<td colspan="2" style="width: 366px;">Use the Addition Property of Equality.</td>
<td style="width: 197px;"><img alt="." src="https://data.ulearngo.com/assets/0fd8dcbf-7259-401e-b8bf-f992699725c2"/></td>
</tr>
<tr>
<td colspan="2" style="width: 366px;">Find the LCD to add the fractions on the right.</td>
<td style="width: 197px;"><img alt="." src="https://data.ulearngo.com/assets/d6060b0f-a56f-4374-9494-e467a1fe2671"/></td>
</tr>
<tr>
<td colspan="2" style="width: 366px;">Simplify</td>
<td style="width: 197px;"><img alt="." src="https://data.ulearngo.com/assets/f7a29b81-a1c2-4867-bb42-c614ff709e63"/></td>
</tr>
<tr>
<td style="width: 171px;">Check:</td>
<td style="width: 195px;"><img alt="." src="https://data.ulearngo.com/assets/332860b8-cbd9-4eec-84ff-4d721fc1a40c"/></td>
<td style="width: 197px;"> </td>
</tr>
<tr>
<td style="width: 171px;"><img alt="." src="https://data.ulearngo.com/assets/7fb488ac-bf82-4e09-b092-1075a189af3b"/></td>
<td style="width: 195px;"><img alt="." src="https://data.ulearngo.com/assets/7942078b-e6fb-4e4a-be30-395cee6ba1d4"/></td>
<td style="width: 197px;"> </td>
</tr>
<tr>
<td style="width: 171px;">Subtract.</td>
<td style="width: 195px;"><img alt="." src="https://data.ulearngo.com/assets/efcb6ce9-1067-4416-915e-533182dc2458"/></td>
<td style="width: 197px;"> </td>
</tr>
<tr>
<td style="width: 171px;">Simplify.</td>
<td style="width: 195px;"><img alt="." src="https://data.ulearngo.com/assets/6337a41f-146e-413a-8963-1d11799addb7"/></td>
<td style="width: 197px;"> </td>
</tr>
<tr>
<td style="width: 171px;">The solution checks.</td>
<td style="width: 195px;"> </td>
<td style="width: 197px;"> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>In <a data-page="32" href="https://app.ulearngo.com/tutorial/mathematics-105/">Mathematics 105</a>, we solved equations that contained decimals. We’ll review this next.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Solve \(a-3.7=4.3.\)</p>
</div>
<div>
<h3>Solution</h3>
<table data-label="" style="width: 645px;" summary="The top line says a minus 3.7 equals 4.3. The next line says “Use the Addition Property of Equality.” Beside that is a minus 3.7 plus a red 3.7 equals 4.3 plus a red 3.7. The next line says to add and shows a equals 8. The next line says to check and shows a minus 3.7 equals 4.3. Below that is “Substitute a equals 8” and a red 8 minus 3.7 followed by an equal sign with a question mark and 4.3. The last line says to simplify and shows 4.3 equals 4.3 with a check mark.">
<tbody>
<tr>
<td colspan="2" style="width: 415px;"> </td>
<td style="width: 214px;"><img alt="." src="https://data.ulearngo.com/assets/ed69112e-a294-49cb-b48d-3d17c45ed50a"/></td>
</tr>
<tr>
<td colspan="2" style="width: 415px;">Use the Addition Property of Equality.</td>
<td style="width: 214px;"><img alt="." src="https://data.ulearngo.com/assets/359cc22e-2bda-4915-acbb-4e5dbf57ba4c"/></td>
</tr>
<tr>
<td colspan="2" style="width: 415px;">Add.</td>
<td style="width: 214px;"><img alt="." src="https://data.ulearngo.com/assets/c10fb25f-74af-4d71-9e65-03a704e59fd4"/></td>
</tr>
<tr>
<td style="width: 199px;">Check:</td>
<td style="width: 216px;"><img alt="." src="https://data.ulearngo.com/assets/d68b59e7-bc5e-48bf-a93d-58a20f21970b"/></td>
<td style="width: 214px;"> </td>
</tr>
<tr>
<td style="width: 199px;">Substitute \(a=8\).</td>
<td style="width: 216px;"><img alt="." src="https://data.ulearngo.com/assets/e82b0ecd-930c-41a7-9bb9-f18496a466de"/></td>
<td style="width: 214px;"> </td>
</tr>
<tr>
<td style="width: 199px;">Simplify.</td>
<td style="width: 216px;"><img alt="." src="https://data.ulearngo.com/assets/6433e599-2c7f-4092-bf02-fdea74db53fa"/></td>
<td style="width: 214px;"> </td>
</tr>
<tr>
<td style="width: 199px;">The solution checks.</td>
<td style="width: 216px;"> </td>
<td style="width: 214px;"> </td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<h3>Optional Video: Solving One Step Equations By Addition and Subtraction</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="337" src="https://www.youtube.com/embed/E2povst-oYs?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="599"><span class="mce_SELRES_start" data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;">﻿</span></iframe></p>

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Solving Equations Using the Subtraction and Addition Properties of Equality

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

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Discover how to solve linear equations using various properties of equality, simplifying equations, translation and solving of equations, and a general strategy approach for equations with fraction and decimal coefficients.


Solving Equations Using the Subtraction and Addition Properties of Equality

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