{"id":16211,"date":"2021-07-04T03:50:40","date_gmt":"2021-07-04T03:50:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/?post_type=chapter&#038;p=16211"},"modified":"2021-07-04T03:50:40","modified_gmt":"2021-07-04T03:50:40","slug":"review-solving-multi-step-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/chapter\/review-solving-multi-step-equations\/","title":{"raw":"REVIEW: Solving Multi-Step Equations","rendered":"REVIEW: Solving Multi-Step Equations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use properties of equality to isolate variables and solve algebraic equations<\/li>\r\n \t<li>Use the distributive property<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Use properties of equality to isolate variables and solve algebraic equations<\/h2>\r\nThere are some <b>equations<\/b> that you can solve in your head quickly. For example\u2014what is the value of <i>y<\/i> in the equation [latex]2y=6[\/latex]? Chances are you didn\u2019t need to get out a pencil and paper to calculate that [latex]y=3[\/latex]. You only needed to do one thing to get the answer: divide 6 by 2.\r\n\r\nOther equations are more complicated. Solving [latex]\\displaystyle 4\\left( \\frac{1}{3}t+\\frac{1}{2}\\right)=6[\/latex] without writing anything down is difficult. That\u2019s because this equation contains not just a <b>variable<\/b> but also fractions and <b>terms<\/b> inside parentheses. This is a <b>multi-step equation<\/b>, one that takes several steps to solve. You've probably studied these equations previously and seen that, although multi-step equations take more time and require more operations, they can still be simplified and solved by applying basic algebraic rules.\r\n\r\nRemember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The <b>addition property of equality<\/b> and the <b>multiplication property of equality<\/b> explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you\u2019ll keep both sides of the equation equal.\r\n\r\nIf the equation is in the form [latex]ax+b=c[\/latex], where <i>x<\/i> is the variable, you can solve the equation as before. First \u201cundo\u201d the addition and subtraction, and then \u201cundo\u201d the multiplication and division.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3y+2=11[\/latex].\r\n\r\n[reveal-answer q=\"843520\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843520\"]\r\n\r\nSubtract 2 from both sides of the equation to get the term with the variable by itself.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,\\,\\,\\,\\,-2}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/p>\r\nDivide both sides of the equation by 3 to get a coefficient of 1 for the variable.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see examples of solving two step linear equations.\r\nhttps:\/\/youtu.be\/fCyxSVQKeRw\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3x+5x+4-x+7=88[\/latex].\r\n\r\n[reveal-answer q=\"455516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455516\"]\r\n\r\nThere are three like terms [latex]3x[\/latex], [latex]5x[\/latex],\u00a0and\u00a0[latex]\u2013x[\/latex]\u00a0involving a variable.\u00a0Combine these like terms.\u00a04 and 7 are also like terms and can be added.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+4+7=\\,\\,\\,88\\end{array}[\/latex]<\/p>\r\nThe equation is now in the form\u00a0[latex]ax+b=c[\/latex], so we can solve as before.\r\n<p style=\"text-align: center;\">[latex]7x+11\\,\\,\\,=\\,\\,\\,88[\/latex]<\/p>\r\nSubtract 11 from both sides.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\r\nDivide both sides by 7.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x=11[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows an example of solving a linear equation that requires combining like terms.\r\n\r\nhttps:\/\/youtu.be\/ez_sP2OTGjU\r\n\r\nSome equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].\r\n\r\nTo solve this equation, we need to \u201cmove\u201d one of the variable terms.\u00a0This can make it difficult to decide which side to work with. It doesn\u2019t matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex], however, to avoid negative coefficients, you can move the smaller term.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nSolve:\u00a0[latex]4x-6=2x+10[\/latex]\r\n\r\n[reveal-answer q=\"457216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"457216\"]\r\n\r\nChoose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\,\\,\\,4x-6=2x+10\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x}\\\\\\,\\,\\,4x-6=10[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now add\u00a06 to both\u00a0sides to isolate the term with the variable.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\4x=16\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now divide each side by 4 to isolate the variable x.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{4x}{4}=\\frac{16}{4}\\\\\\\\x=4\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this video, you can see an example of solving equations that have variables on both sides of the equal sign.\r\nhttps:\/\/youtu.be\/f3ujWNPL0Bw\r\n<h2 id=\"title2\">The Distributive Property<\/h2>\r\nAs we solve linear equations, we often need to do some work to write\u00a0the linear equations in a form we are familiar with solving.\u00a0This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.\r\n\r\nParentheses can\u00a0make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.\r\n<div class=\"textbox shaded\">\r\n<h3>The Distributive Property of Multiplication<\/h3>\r\nFor all real numbers <i>a, b,<\/i> and <i>c<\/i>,\u00a0[latex]a(b+c)=ab+ac[\/latex].\r\n\r\nWhat this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced\u00a0to <b>isolate the variable<\/b>\u00a0and solve the equation.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]a[\/latex]. [latex]4\\left(2a+3\\right)=28[\/latex]\r\n\r\n[reveal-answer q=\"372387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"372387\"]\r\n\r\nApply the distributive property to expand [latex]4\\left(2a+3\\right)[\/latex] to [latex]8a+12[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\r\nSubtract 12\u00a0from both sides to isolate\u00a0the variable term.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\r\nDivide both terms by 8 to get a coefficient of 1.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]a=2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, see another example of how to use the distributive property to solve a multi-step linear equation.\r\n\r\nhttps:\/\/youtu.be\/aQOkD8L57V0\r\n\r\nIn the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]t[\/latex].\u00a0\u00a0[latex]2\\left(4t-5\\right)=-3\\left(2t+1\\right)[\/latex]\r\n\r\n[reveal-answer q=\"302387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"302387\"]\r\n\r\nApply the distributive property to expand [latex]2\\left(4t-5\\right)[\/latex] to [latex]8t-10[\/latex] and [latex]-3\\left(2t+1\\right)[\/latex] to[latex]-6t-3[\/latex]. Be careful in this step\u2014you are distributing a negative number, so keep track of the sign of each number after you multiply.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\left(4t-5\\right)=-3\\left(2t+1\\right)\\,\\,\\,\\,\\,\\, \\\\ 8t-10=-6t-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd [latex]-6t[\/latex] to both sides to begin combining like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides of the equation to isolate <em>t<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}14t-10=-3\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 14t=\\,\\,\\,7\\,\\end{array}[\/latex]<\/p>\r\nThe last step is to divide both sides by 14 to completely isolate <em>t<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]t=\\frac{1}{2}[\/latex]\r\n\r\nWe simplified the fraction [latex]\\frac{7}{14}[\/latex] into [latex]\\frac{1}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we solve another multi-step equation with two sets of parentheses.\r\n\r\nhttps:\/\/youtu.be\/StomYTb7Xb8","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use properties of equality to isolate variables and solve algebraic equations<\/li>\n<li>Use the distributive property<\/li>\n<\/ul>\n<\/div>\n<h2>Use properties of equality to isolate variables and solve algebraic equations<\/h2>\n<p>There are some <b>equations<\/b> that you can solve in your head quickly. For example\u2014what is the value of <i>y<\/i> in the equation [latex]2y=6[\/latex]? Chances are you didn\u2019t need to get out a pencil and paper to calculate that [latex]y=3[\/latex]. You only needed to do one thing to get the answer: divide 6 by 2.<\/p>\n<p>Other equations are more complicated. Solving [latex]\\displaystyle 4\\left( \\frac{1}{3}t+\\frac{1}{2}\\right)=6[\/latex] without writing anything down is difficult. That\u2019s because this equation contains not just a <b>variable<\/b> but also fractions and <b>terms<\/b> inside parentheses. This is a <b>multi-step equation<\/b>, one that takes several steps to solve. You&#8217;ve probably studied these equations previously and seen that, although multi-step equations take more time and require more operations, they can still be simplified and solved by applying basic algebraic rules.<\/p>\n<p>Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The <b>addition property of equality<\/b> and the <b>multiplication property of equality<\/b> explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you\u2019ll keep both sides of the equation equal.<\/p>\n<p>If the equation is in the form [latex]ax+b=c[\/latex], where <i>x<\/i> is the variable, you can solve the equation as before. First \u201cundo\u201d the addition and subtraction, and then \u201cundo\u201d the multiplication and division.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]3y+2=11[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843520\">Show Solution<\/span><\/p>\n<div id=\"q843520\" class=\"hidden-answer\" style=\"display: none\">\n<p>Subtract 2 from both sides of the equation to get the term with the variable by itself.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,\\,\\,\\,\\,-2}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/p>\n<p>Divide both sides of the equation by 3 to get a coefficient of 1 for the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see examples of solving two step linear equations.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solving Two Step Equations (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fCyxSVQKeRw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]3x+5x+4-x+7=88[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455516\">Show Solution<\/span><\/p>\n<div id=\"q455516\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are three like terms [latex]3x[\/latex], [latex]5x[\/latex],\u00a0and\u00a0[latex]\u2013x[\/latex]\u00a0involving a variable.\u00a0Combine these like terms.\u00a04 and 7 are also like terms and can be added.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+4+7=\\,\\,\\,88\\end{array}[\/latex]<\/p>\n<p>The equation is now in the form\u00a0[latex]ax+b=c[\/latex], so we can solve as before.<\/p>\n<p style=\"text-align: center;\">[latex]7x+11\\,\\,\\,=\\,\\,\\,88[\/latex]<\/p>\n<p>Subtract 11 from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\n<p>Divide both sides by 7.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=11[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following video shows an example of solving a linear equation that requires combining like terms.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solving an Equation that Requires Combining Like Terms\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ez_sP2OTGjU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].<\/p>\n<p>To solve this equation, we need to \u201cmove\u201d one of the variable terms.\u00a0This can make it difficult to decide which side to work with. It doesn\u2019t matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex], however, to avoid negative coefficients, you can move the smaller term.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Solve:\u00a0[latex]4x-6=2x+10[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q457216\">Show Solution<\/span><\/p>\n<div id=\"q457216\" class=\"hidden-answer\" style=\"display: none\">\n<p>Choose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\,\\,\\,4x-6=2x+10\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x}\\\\\\,\\,\\,4x-6=10[\/latex]<\/p>\n<p style=\"text-align: left;\">Now add\u00a06 to both\u00a0sides to isolate the term with the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\4x=16\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Now divide each side by 4 to isolate the variable x.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{4x}{4}=\\frac{16}{4}\\\\\\\\x=4\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this video, you can see an example of solving equations that have variables on both sides of the equal sign.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Solve an Equation with Variable on Both Sides\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/f3ujWNPL0Bw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">The Distributive Property<\/h2>\n<p>As we solve linear equations, we often need to do some work to write\u00a0the linear equations in a form we are familiar with solving.\u00a0This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.<\/p>\n<p>Parentheses can\u00a0make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Distributive Property of Multiplication<\/h3>\n<p>For all real numbers <i>a, b,<\/i> and <i>c<\/i>,\u00a0[latex]a(b+c)=ab+ac[\/latex].<\/p>\n<p>What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced\u00a0to <b>isolate the variable<\/b>\u00a0and solve the equation.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]a[\/latex]. [latex]4\\left(2a+3\\right)=28[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q372387\">Show Solution<\/span><\/p>\n<div id=\"q372387\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the distributive property to expand [latex]4\\left(2a+3\\right)[\/latex] to [latex]8a+12[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\n<p>Subtract 12\u00a0from both sides to isolate\u00a0the variable term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\n<p>Divide both terms by 8 to get a coefficient of 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]a=2[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, see another example of how to use the distributive property to solve a multi-step linear equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Solving an Equation with One Set of Parentheses\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aQOkD8L57V0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]t[\/latex].\u00a0\u00a0[latex]2\\left(4t-5\\right)=-3\\left(2t+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q302387\">Show Solution<\/span><\/p>\n<div id=\"q302387\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the distributive property to expand [latex]2\\left(4t-5\\right)[\/latex] to [latex]8t-10[\/latex] and [latex]-3\\left(2t+1\\right)[\/latex] to[latex]-6t-3[\/latex]. Be careful in this step\u2014you are distributing a negative number, so keep track of the sign of each number after you multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\left(4t-5\\right)=-3\\left(2t+1\\right)\\,\\,\\,\\,\\,\\, \\\\ 8t-10=-6t-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add [latex]-6t[\/latex] to both sides to begin combining like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add 10 to both sides of the equation to isolate <em>t<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}14t-10=-3\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 14t=\\,\\,\\,7\\,\\end{array}[\/latex]<\/p>\n<p>The last step is to divide both sides by 14 to completely isolate <em>t<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]t=\\frac{1}{2}[\/latex]<\/p>\n<p>We simplified the fraction [latex]\\frac{7}{14}[\/latex] into [latex]\\frac{1}{2}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the following video, we solve another multi-step equation with two sets of parentheses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Solving an Equation with Parentheses on Both Sides\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/StomYTb7Xb8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"author":407919,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16211","chapter","type-chapter","status-publish","hentry"],"part":16055,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/16211","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/users\/407919"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/16211\/revisions"}],"predecessor-version":[{"id":16212,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/16211\/revisions\/16212"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/parts\/16055"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/16211\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/media?parent=16211"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapter-type?post=16211"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/contributor?post=16211"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/license?post=16211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}