{"id":4528,"date":"2017-06-07T18:50:29","date_gmt":"2017-06-07T18:50:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiplication-property-of-equality\/"},"modified":"2017-08-15T12:19:17","modified_gmt":"2017-08-15T12:19:17","slug":"read-multiplication-property-of-equality","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiplication-property-of-equality\/","title":{"raw":"Multiplication Property of Equality","rendered":"Multiplication Property of Equality"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use the multiplication property of equality\r\n<ul>\r\n \t<li>Solve algebraic equations using the multiplication property of equality<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title2\">Solve algebraic equations using the multiplication property of equality<\/h2>\r\nJust as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let\u2019s look at a numeric equation, [latex]5\\cdot3=15[\/latex], to start. If you multiply both sides of this equation by 2, you will still have a true equation.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5\\cdot 3=15\\,\\,\\,\\,\\,\\,\\, \\\\ 5\\cdot3\\cdot2=15\\cdot2 \\\\ 30=30\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThis characteristic of equations is generalized in the <b>multiplication property of equality<\/b>.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication Property of Equality<\/h3>\r\nFor all real numbers <i>a<\/i>, <i>b<\/i>, and <i>c<\/i>: If <i>a<\/i> = <i>b<\/i>, then [latex]a\\cdot{c}=b\\cdot{c}[\/latex]\u00a0(or <i>ab<\/i> = <i>ac<\/i>).\r\n\r\nIf two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.\r\n\r\n<\/div>\r\nWhen the equation involves multiplication or division, you can \u201cundo\u201d these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3x=24[\/latex]. When you are done, check your solution.\r\n\r\n[reveal-answer q=\"42404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42404\"]Divide both sides of the equation by 3 to isolate the variable (have a coefficient of 1).\u00a0Dividing by 3 is the same as having multiplied by [latex] \\frac{1}{3}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{3x}=\\underline{24}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\x=8\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck by substituting your solution, 8, for the variable in the original equation.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x=24 \\\\ 3\\cdot8=24 \\\\ 24=24\\end{array}[\/latex]<\/p>\r\nThe solution is correct!\r\n<h4>Answer<\/h4>\r\n[latex]x=8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can also multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex] \\frac{1}{2 }{ x }={ 8}[\/latex] for x.\r\n[reveal-answer q=\"128018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"128018\"]The only difference between this and the previous example is that the coefficient on <em>x<\/em> is a fraction. In the last example we used the reciprocal of 3 to isolate the <em>x<\/em> (disguised as division). Now we can use the reciprocal of [latex]\\frac{1}{2}[\/latex], which is 2.\r\n<p style=\"text-align: left\">Multiply both sides by 2:<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\left(2\\right)\\frac{1}{2 }{ x }=\\left(2\\right){ 8}\\\\\\left(2\\right)\\frac{1}{2 } = 1\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\left(2\\right)8 = 16\\,\\,\\,\\,\\,\\\\{ x }=16\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the video below you will see examples of how to use the multiplication property of equality to solve one-step equations with integers and fractions.\r\nhttps:\/\/youtu.be\/BN7iVWWl2y0\r\n\r\nIn the next example, we will solve a one-step equation using the multiplication property of equality. You will see that the variable is part of a fraction in the given equation, and using the multiplication property of equality allows us to remove the variable from the fraction. Remember that fractions imply division, so you can think of this as the variable <em>k<\/em> is being divided by 10. To \"undo\" the division, you can use multiplication to isolate <em>k<\/em>. Lastly, note that there is a negative term in the equation, so it will be important to think about the sign of each term as you work through the problem. Stop after each step you take to make sure all the terms have the correct sign.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]-\\frac{7}{2}=\\frac{k}{10}[\/latex] for <em>k<\/em>.\r\n\r\n[reveal-answer q=\"471772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"471772\"]We want to isolate the <em>k<\/em>, which is being divided by 10. The first thing we should do is multiply both sides by 10.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{ multiply the left side by 10 }\\\\\\\\\\left(10\\right)-\\frac{7}{2}=\\frac{10\\cdot7}{2} = \\frac{70}{2} = 35\\\\\\\\\\text{ now multiply the right side by 10 }\\\\\\\\\\,\\,\\,\\frac{k}{10}\\left(10\\right) = \\frac{k\\cdot10}{10} = k\\\\\\\\\\text{ now replace your results in the equation }\\\\35=k\\end{array}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left\">Answer<\/h4>\r\nWe write the k on the left side as a matter of convention.\r\n\r\n[latex]k=35[\/latex]\r\n<p style=\"text-align: center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the following video you will see examples of using the multiplication property of equality to solve a one-step equation involving negative fractions.\r\nhttps:\/\/youtu.be\/AhBdGeUGgsI\r\n\r\n&nbsp;\r\n<h2>Summary<\/h2>\r\nEquations are mathematical statements that combine two expressions of equal value. An algebraic equation can be solved by isolating the variable on one side of the equation using the properties of equality. To check the solution of an algebraic equation, substitute the value of the variable into the original equation.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use the multiplication property of equality\n<ul>\n<li>Solve algebraic equations using the multiplication property of equality<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title2\">Solve algebraic equations using the multiplication property of equality<\/h2>\n<p>Just as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let\u2019s look at a numeric equation, [latex]5\\cdot3=15[\/latex], to start. If you multiply both sides of this equation by 2, you will still have a true equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5\\cdot 3=15\\,\\,\\,\\,\\,\\,\\, \\\\ 5\\cdot3\\cdot2=15\\cdot2 \\\\ 30=30\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>This characteristic of equations is generalized in the <b>multiplication property of equality<\/b>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplication Property of Equality<\/h3>\n<p>For all real numbers <i>a<\/i>, <i>b<\/i>, and <i>c<\/i>: If <i>a<\/i> = <i>b<\/i>, then [latex]a\\cdot{c}=b\\cdot{c}[\/latex]\u00a0(or <i>ab<\/i> = <i>ac<\/i>).<\/p>\n<p>If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.<\/p>\n<\/div>\n<p>When the equation involves multiplication or division, you can \u201cundo\u201d these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]3x=24[\/latex]. When you are done, check your solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q42404\">Show Solution<\/span><\/p>\n<div id=\"q42404\" class=\"hidden-answer\" style=\"display: none\">Divide both sides of the equation by 3 to isolate the variable (have a coefficient of 1).\u00a0Dividing by 3 is the same as having multiplied by [latex]\\frac{1}{3}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{3x}=\\underline{24}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\x=8\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Check by substituting your solution, 8, for the variable in the original equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x=24 \\\\ 3\\cdot8=24 \\\\ 24=24\\end{array}[\/latex]<\/p>\n<p>The solution is correct!<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can also multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]\\frac{1}{2 }{ x }={ 8}[\/latex] for x.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q128018\">Show Solution<\/span><\/p>\n<div id=\"q128018\" class=\"hidden-answer\" style=\"display: none\">The only difference between this and the previous example is that the coefficient on <em>x<\/em> is a fraction. In the last example we used the reciprocal of 3 to isolate the <em>x<\/em> (disguised as division). Now we can use the reciprocal of [latex]\\frac{1}{2}[\/latex], which is 2.<\/p>\n<p style=\"text-align: left\">Multiply both sides by 2:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\left(2\\right)\\frac{1}{2 }{ x }=\\left(2\\right){ 8}\\\\\\left(2\\right)\\frac{1}{2 } = 1\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\left(2\\right)8 = 16\\,\\,\\,\\,\\,\\\\{ x }=16\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p style=\"text-align: center\"><\/div>\n<\/div>\n<\/div>\n<p>In the video below you will see examples of how to use the multiplication property of equality to solve one-step equations with integers and fractions.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solving One Step Equations Using Multiplication and Division (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BN7iVWWl2y0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, we will solve a one-step equation using the multiplication property of equality. You will see that the variable is part of a fraction in the given equation, and using the multiplication property of equality allows us to remove the variable from the fraction. Remember that fractions imply division, so you can think of this as the variable <em>k<\/em> is being divided by 10. To &#8220;undo&#8221; the division, you can use multiplication to isolate <em>k<\/em>. Lastly, note that there is a negative term in the equation, so it will be important to think about the sign of each term as you work through the problem. Stop after each step you take to make sure all the terms have the correct sign.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]-\\frac{7}{2}=\\frac{k}{10}[\/latex] for <em>k<\/em>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q471772\">Show Solution<\/span><\/p>\n<div id=\"q471772\" class=\"hidden-answer\" style=\"display: none\">We want to isolate the <em>k<\/em>, which is being divided by 10. The first thing we should do is multiply both sides by 10.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{ multiply the left side by 10 }\\\\\\\\\\left(10\\right)-\\frac{7}{2}=\\frac{10\\cdot7}{2} = \\frac{70}{2} = 35\\\\\\\\\\text{ now multiply the right side by 10 }\\\\\\\\\\,\\,\\,\\frac{k}{10}\\left(10\\right) = \\frac{k\\cdot10}{10} = k\\\\\\\\\\text{ now replace your results in the equation }\\\\35=k\\end{array}[\/latex]<\/p>\n<h4 style=\"text-align: left\">Answer<\/h4>\n<p>We write the k on the left side as a matter of convention.<\/p>\n<p>[latex]k=35[\/latex]<\/p>\n<p style=\"text-align: center\"><\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see examples of using the multiplication property of equality to solve a one-step equation involving negative fractions.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solving One Step Equations Using Multiplication (Fractions)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/AhBdGeUGgsI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>Equations are mathematical statements that combine two expressions of equal value. An algebraic equation can be solved by isolating the variable on one side of the equation using the properties of equality. To check the solution of an algebraic equation, substitute the value of the variable into the original equation.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4528\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Solving One Step Equations Using Multiplication and Division (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BN7iVWWl2y0\">https:\/\/youtu.be\/BN7iVWWl2y0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving Absolute Value Equation Using Multiplication and Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CTLnJ955xzc\">https:\/\/youtu.be\/CTLnJ955xzc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Solving One Step Equations Using Multiplication (Fractions). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/AhBdGeUGgsI\">https:\/\/youtu.be\/AhBdGeUGgsI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Solving One Step Equations Using Multiplication and Division (Basic)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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