{"id":4422,"date":"2020-04-13T13:02:00","date_gmt":"2020-04-13T13:02:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4422"},"modified":"2021-02-06T00:04:58","modified_gmt":"2021-02-06T00:04:58","slug":"solving-multi-step-equations-using-a-general-strategy","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/solving-multi-step-equations-using-a-general-strategy\/","title":{"raw":"Solving Multi-Step Equations Using a General Strategy","rendered":"Solving Multi-Step Equations Using a General Strategy"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the steps of a general problem solving strategy for solving linear equations<\/li>\r\n \t<li>Use a general problem solving strategy to solve linear equations that require several steps<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this section, we will lay out an overall strategy that can be used to solve <em>any<\/em> linear equation. We call this the <em>general strategy<\/em>. Some equations won\u2019t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\">general strategy for solving linear equations<\/h3>\r\n<ol id=\"eip-id1168467248588\" class=\"stepwise\">\r\n \t<li>Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.<\/li>\r\n \t<li>If there are fractions or decimals in the equation, multiply by the least common denominator to clear them.<\/li>\r\n \t<li>Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Make the coefficient of the variable term to equal to [latex]1[\/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.<\/li>\r\n \t<li>Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve: [latex]3\\left(x+2\\right)=18[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468387403\" class=\"unnumbered unstyled\" summary=\"Simplify each side of the equation as much as possible. Use the Distributive Property.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3(x+2)=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each side of the equation as much as possible.Use the Distributive Property.<\/td>\r\n<td>[latex]3x+6=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Collect all variable terms on one side of the equation\u2014all [latex]x[\/latex] s are already on the left side.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Collect constant terms on the other side of the equation.Subtract [latex]6[\/latex] from each side.<\/td>\r\n<td>[latex]3x+6\\color{red}{-6}=18\\color{red}{-6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]3x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Make the coefficient of the variable term equal to [latex]1[\/latex]. Divide each side by [latex]3[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{3x}{\\color{red}{3}}\\normalsize =\\Large\\frac{12}{\\color{red}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>\u00a0[latex]3(x+2)=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]x=4[\/latex].<\/td>\r\n<td>[latex]3(\\color{red}{4}+2)\\stackrel{\\text{?}}{=}18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3(6)\\stackrel{\\text{?}}{=}18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18=18\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1818&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve: [latex]-\\left(x+5\\right)=7[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"190834\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168469659755\" class=\"unnumbered unstyled\" summary=\"Simplify each side of the equation as much as possible by distributing. The only x term is on the left side, so all variable terms are on the left side of the equation.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-(x+5)=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each side of the equation as much as possible by distributing.The only [latex]x[\/latex] term is on the left side, so all variable terms are on the left side of the equation.<\/td>\r\n<td>[latex]-x-5=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]5[\/latex] to both sides to get all constant terms on the right side of the equation.<\/td>\r\n<td>[latex]-x-5\\color{red}{+5}=7\\color{red}{+5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Make the coefficient of the variable term equal to [latex]1[\/latex] by multiplying both sides by [latex]-1[\/latex].<\/td>\r\n<td>[latex]\\color{red}{-1}(-x)=\\color{red}{-1}(12)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=-12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>\u00a0[latex]-(x+5)=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]x=-12[\/latex].<\/td>\r\n<td>\u00a0[latex]-(\\color{red}{-12}+5)\\stackrel{\\text{?}}{=}7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]-(-7)\\stackrel{\\text{?}}{=}7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7=7\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142578&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve: [latex]4\\left(x - 2\\right)+5=-3[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"743652\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"743652\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168469612809\" class=\"unnumbered unstyled\" summary=\"The top line shows 4 times parentheses x minus 2 plus 5 equals negative 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4(x-2)+5=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each side of the equation as much as possible.Distribute.<\/td>\r\n<td>[latex]4x-8+5=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms<\/td>\r\n<td>[latex]4x-3=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The only [latex]x[\/latex] is on the left side, so all variable terms are on one side of the equation.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]3[\/latex] to both sides to get all constant terms on the other side of the equation.<\/td>\r\n<td>[latex]4x-3\\color{red}{+3}=-3\\color{red}{+3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]4x=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Make the coefficient of the variable term equal to [latex]1[\/latex] by dividing both sides by [latex]4[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{4x}{\\color{red}{4}}\\normalsize =\\Large\\frac{0}{\\color{red}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>\u00a0[latex]4(x-2)+5=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]x=0[\/latex].<\/td>\r\n<td>[latex]4(\\color{red}{0-2})+5\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4(-2)+5\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]-8+5\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]-3=-3\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142579&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve: [latex]8 - 2\\left(3y+5\\right)=0[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"822344\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"822344\"]<\/p>\r\nSolution:\r\nBe careful when distributing the negative.\r\n<table id=\"eip-id1168467174932\" class=\"unnumbered unstyled\" summary=\"The top line says 8 minus 2 parentheses 3y plus 5 equals 0. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8-2(3y+5)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify\u2014use the Distributive Property.<\/td>\r\n<td>[latex]8-6y-10=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]-6y-2=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]2[\/latex] to both sides to collect constants on the right.<\/td>\r\n<td>[latex]-6y-2\\color{red}{+2}=0\\color{red}{+2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-6y=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by [latex]-6[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{-6y}{\\color{red}{-6}}\\normalsize =\\Large\\frac{2}{\\color{red}{-6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=-\\Large\\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>\u00a0[latex]8-2(3y+5)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]y=-\\frac{1}{3}[\/latex]<\/td>\r\n<td>[latex]8-2[3(\\color{red}{-\\Large\\frac{1}{3}}\\normalsize )+5]\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8-2(-1+5)\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8-2(4)\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]8-8\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0=0\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142580&amp;theme=oea&amp;iframe_resize_id=mom4[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n&nbsp;\r\n\r\nSolve: [latex]3\\left(x - 2\\right)-5=4\\left(2x+1\\right)+5[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"123432\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"123432\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168468704060\" class=\"unnumbered unstyled\" summary=\"The top line says 3 parentheses x minus 2 minus 5 equals 4 parentheses 2x plus 1 plus 5. The next line says, \">\r\n<tbody>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\"><\/td>\r\n<td style=\"height: 14px\">[latex]3(x-2)-5=4(2x+1)+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Distribute.<\/td>\r\n<td style=\"height: 14px\">[latex]3x-6-5=8x+4+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Combine like terms.<\/td>\r\n<td style=\"height: 14px\">[latex]3x-11=8x+9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Subtract [latex]3x[\/latex] to get all the variables on the right since [latex]8&gt;3[\/latex] .<\/td>\r\n<td style=\"height: 14px\">[latex]3x\\color{red}{-3x}-11=8x\\color{red}{-3x}+9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Simplify.<\/td>\r\n<td style=\"height: 14px\">[latex]-11=5x+9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Subtract [latex]9[\/latex] to get the constants on the left.<\/td>\r\n<td style=\"height: 14px\">[latex]-11\\color{red}{-9}=5x+9\\color{red}{-9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14.5557px\">\r\n<td style=\"height: 14.5557px\">Simplify.<\/td>\r\n<td style=\"height: 14.5557px\">[latex]-20=5x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Divide by [latex]5[\/latex].<\/td>\r\n<td style=\"height: 14px\">[latex]\\Large\\frac{-20}{\\color{red}{5}}\\normalsize =\\Large\\frac{5x}{\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Simplify.<\/td>\r\n<td style=\"height: 14px\">[latex]-4=x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\">Check: Substitute: [latex]-4=x[\/latex] .<\/td>\r\n<td style=\"height: 14px\">\u00a0[latex]3(\\color{red}{-4}-2)-5\\overset{?}{=}4(2(\\color{red}{-4})+1)+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\"><\/td>\r\n<td style=\"height: 14px\">[latex]3(-6)-5\\overset{?}{=}4(-8+1)+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\"><\/td>\r\n<td style=\"height: 14px\">[latex]-18-5\\overset{?}{=}4(-7)+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\"><\/td>\r\n<td style=\"height: 14px\">[latex]-23\\overset{?}{=}-28+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"height: 14px\"><\/td>\r\n<td style=\"height: 14px\">[latex]-23\\overset{?}{=}-23\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142586&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142587&amp;theme=oea&amp;iframe_resize_id=mom6[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve: [latex]\\Large\\frac{1}{2}\\normalsize\\left(6x - 2\\right)=5-x[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"735621\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"735621\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168469851853\" class=\"unnumbered unstyled\" summary=\"The top line says one-half times parentheses 6x minus 2 equals 5 minus x. The next line says, \">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{2}\\normalsize(6x-2)=5-x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Distribute.<\/td>\r\n<td style=\"height: 15px\">[latex]3x-1=5-x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Add [latex]x[\/latex] to get all the variables on the left.<\/td>\r\n<td style=\"height: 15px\">[latex]3x-1\\color{red}{+x}=5-x\\color{red}{+x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Simplify.<\/td>\r\n<td style=\"height: 15px\">[latex]4x-1=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Add [latex]1[\/latex] to get constants on the right.<\/td>\r\n<td style=\"height: 15px\">[latex]4x-1\\color{red}{+1}=5\\color{red}{+1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Simplify.<\/td>\r\n<td style=\"height: 15px\">[latex]4x=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Divide by [latex]4[\/latex].<\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{4x}{\\color{red}{4}}\\normalsize =\\Large\\frac{6}{\\color{red}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Simplify.<\/td>\r\n<td style=\"height: 15px\">[latex]x=\\Large\\frac{3}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.8281px\">\r\n<td style=\"height: 15.8281px\">Check: Let [latex]x=\\Large\\frac{3}{2}[\/latex] .<\/td>\r\n<td style=\"height: 15.8281px\">\u00a0[latex]\\Large\\frac{1}{2}\\normalsize (6(\\Large\\frac{\\color{red}{3}}{\\color{red}{2}}\\normalsize )-2)\\overset{?}{=}5-(\\Large\\frac{\\color{red}3}{\\color{red}2})[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{2}\\normalsize(9-2)\\overset{?}{=}\\Large\\frac{10}{2}\\normalsize -\\Large\\frac{3}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{2}\\normalsize(7)\\overset{?}{=}\\Large\\frac{7}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{7}{2}\\normalsize =\\Large\\frac{7}{2}\\normalsize\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142589&amp;theme=oea&amp;iframe_resize_id=mom7[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142581&amp;theme=oea&amp;iframe_resize_id=mom8[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142582&amp;theme=oea&amp;iframe_resize_id=mom9[\/embed]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nWatch the following video to see another example of how to solve an equation that requires distributing a fraction.\r\n\r\nhttps:\/\/youtu.be\/1dmEoG7DkN4\r\n\r\nIn the next video example we show an example of solving an equation that requires distributing a fraction. \u00a0In this case, you will need to clear fractions after you distribute.\r\n\r\nhttps:\/\/youtu.be\/-P4KZECxo8Y\r\n\r\nIn many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"964378\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"964378\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168468569426\" class=\"unnumbered unstyled\" summary=\"The top line says 0.24 times parentheses 100x plus 5 equals 0.4 times parentheses 30x plus 15. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]0.45a+0.36=0.3a+0.66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply by the least common denominator, 100<\/td>\r\n<td>[latex]45a+36=30a+66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]30a[\/latex] to get all the [latex]x[\/latex] s to the left.<\/td>\r\n<td>[latex]45a\\color{red}{-30a}+36=30a+66\\color{red}{-30a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]15a+36=66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]36[\/latex] to get the constants to the right.<\/td>\r\n<td>[latex]15a+36\\color{red}{-36}=66\\color{red}{-36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]15a=30[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]\\large{\\frac{15a}{15}=\\frac{30}{15}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x = 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check: Let [latex]x=2[\/latex]<\/td>\r\n<td>\u00a0[latex]0.45\\left(2+0.8\\right)=0.3\\left(2+2.2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]1.26=1.26\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]140292[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video provides another example of how to solve an equation that requires distributing a decimal.\r\n\r\nhttps:\/\/youtu.be\/k0K8mat_EaI","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the steps of a general problem solving strategy for solving linear equations<\/li>\n<li>Use a general problem solving strategy to solve linear equations that require several steps<\/li>\n<\/ul>\n<\/div>\n<p>In this section, we will lay out an overall strategy that can be used to solve <em>any<\/em> linear equation. We call this the <em>general strategy<\/em>. Some equations won\u2019t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">general strategy for solving linear equations<\/h3>\n<ol id=\"eip-id1168467248588\" class=\"stepwise\">\n<li>Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.<\/li>\n<li>If there are fractions or decimals in the equation, multiply by the least common denominator to clear them.<\/li>\n<li>Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Make the coefficient of the variable term to equal to [latex]1[\/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.<\/li>\n<li>Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve: [latex]3\\left(x+2\\right)=18[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168468387403\" class=\"unnumbered unstyled\" summary=\"Simplify each side of the equation as much as possible. Use the Distributive Property.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3(x+2)=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each side of the equation as much as possible.Use the Distributive Property.<\/td>\n<td>[latex]3x+6=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Collect all variable terms on one side of the equation\u2014all [latex]x[\/latex] s are already on the left side.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Collect constant terms on the other side of the equation.Subtract [latex]6[\/latex] from each side.<\/td>\n<td>[latex]3x+6\\color{red}{-6}=18\\color{red}{-6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]3x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Make the coefficient of the variable term equal to [latex]1[\/latex]. Divide each side by [latex]3[\/latex].<\/td>\n<td>[latex]\\Large\\frac{3x}{\\color{red}{3}}\\normalsize =\\Large\\frac{12}{\\color{red}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>\u00a0[latex]3(x+2)=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]x=4[\/latex].<\/td>\n<td>[latex]3(\\color{red}{4}+2)\\stackrel{\\text{?}}{=}18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3(6)\\stackrel{\\text{?}}{=}18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18=18\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1818\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1818&#38;theme=oea&#38;iframe_resize_id=ohm1818&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve: [latex]-\\left(x+5\\right)=7[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469659755\" class=\"unnumbered unstyled\" summary=\"Simplify each side of the equation as much as possible by distributing. The only x term is on the left side, so all variable terms are on the left side of the equation.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-(x+5)=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each side of the equation as much as possible by distributing.The only [latex]x[\/latex] term is on the left side, so all variable terms are on the left side of the equation.<\/td>\n<td>[latex]-x-5=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]5[\/latex] to both sides to get all constant terms on the right side of the equation.<\/td>\n<td>[latex]-x-5\\color{red}{+5}=7\\color{red}{+5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Make the coefficient of the variable term equal to [latex]1[\/latex] by multiplying both sides by [latex]-1[\/latex].<\/td>\n<td>[latex]\\color{red}{-1}(-x)=\\color{red}{-1}(12)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=-12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>\u00a0[latex]-(x+5)=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]x=-12[\/latex].<\/td>\n<td>\u00a0[latex]-(\\color{red}{-12}+5)\\stackrel{\\text{?}}{=}7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]-(-7)\\stackrel{\\text{?}}{=}7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]7=7\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142578\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142578&#38;theme=oea&#38;iframe_resize_id=ohm142578&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve: [latex]4\\left(x - 2\\right)+5=-3[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q743652\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q743652\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469612809\" class=\"unnumbered unstyled\" summary=\"The top line shows 4 times parentheses x minus 2 plus 5 equals negative 3. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]4(x-2)+5=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each side of the equation as much as possible.Distribute.<\/td>\n<td>[latex]4x-8+5=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms<\/td>\n<td>[latex]4x-3=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The only [latex]x[\/latex] is on the left side, so all variable terms are on one side of the equation.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]3[\/latex] to both sides to get all constant terms on the other side of the equation.<\/td>\n<td>[latex]4x-3\\color{red}{+3}=-3\\color{red}{+3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]4x=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Make the coefficient of the variable term equal to [latex]1[\/latex] by dividing both sides by [latex]4[\/latex].<\/td>\n<td>[latex]\\Large\\frac{4x}{\\color{red}{4}}\\normalsize =\\Large\\frac{0}{\\color{red}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>\u00a0[latex]4(x-2)+5=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]x=0[\/latex].<\/td>\n<td>[latex]4(\\color{red}{0-2})+5\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4(-2)+5\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]-8+5\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]-3=-3\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142579\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142579&#38;theme=oea&#38;iframe_resize_id=ohm142579&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve: [latex]8 - 2\\left(3y+5\\right)=0[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q822344\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q822344\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nBe careful when distributing the negative.<\/p>\n<table id=\"eip-id1168467174932\" class=\"unnumbered unstyled\" summary=\"The top line says 8 minus 2 parentheses 3y plus 5 equals 0. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]8-2(3y+5)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify\u2014use the Distributive Property.<\/td>\n<td>[latex]8-6y-10=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]-6y-2=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]2[\/latex] to both sides to collect constants on the right.<\/td>\n<td>[latex]-6y-2\\color{red}{+2}=0\\color{red}{+2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-6y=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by [latex]-6[\/latex].<\/td>\n<td>[latex]\\Large\\frac{-6y}{\\color{red}{-6}}\\normalsize =\\Large\\frac{2}{\\color{red}{-6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=-\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>\u00a0[latex]8-2(3y+5)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]y=-\\frac{1}{3}[\/latex]<\/td>\n<td>[latex]8-2[3(\\color{red}{-\\Large\\frac{1}{3}}\\normalsize )+5]\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8-2(-1+5)\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8-2(4)\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]8-8\\stackrel{\\text{?}}{=}0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0=0\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142580\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142580&#38;theme=oea&#38;iframe_resize_id=ohm142580&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>&nbsp;<\/p>\n<p>Solve: [latex]3\\left(x - 2\\right)-5=4\\left(2x+1\\right)+5[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q123432\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q123432\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468704060\" class=\"unnumbered unstyled\" summary=\"The top line says 3 parentheses x minus 2 minus 5 equals 4 parentheses 2x plus 1 plus 5. The next line says,\">\n<tbody>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\"><\/td>\n<td style=\"height: 14px\">[latex]3(x-2)-5=4(2x+1)+5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Distribute.<\/td>\n<td style=\"height: 14px\">[latex]3x-6-5=8x+4+5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Combine like terms.<\/td>\n<td style=\"height: 14px\">[latex]3x-11=8x+9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Subtract [latex]3x[\/latex] to get all the variables on the right since [latex]8>3[\/latex] .<\/td>\n<td style=\"height: 14px\">[latex]3x\\color{red}{-3x}-11=8x\\color{red}{-3x}+9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Simplify.<\/td>\n<td style=\"height: 14px\">[latex]-11=5x+9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Subtract [latex]9[\/latex] to get the constants on the left.<\/td>\n<td style=\"height: 14px\">[latex]-11\\color{red}{-9}=5x+9\\color{red}{-9}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14.5557px\">\n<td style=\"height: 14.5557px\">Simplify.<\/td>\n<td style=\"height: 14.5557px\">[latex]-20=5x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Divide by [latex]5[\/latex].<\/td>\n<td style=\"height: 14px\">[latex]\\Large\\frac{-20}{\\color{red}{5}}\\normalsize =\\Large\\frac{5x}{\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Simplify.<\/td>\n<td style=\"height: 14px\">[latex]-4=x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\">Check: Substitute: [latex]-4=x[\/latex] .<\/td>\n<td style=\"height: 14px\">\u00a0[latex]3(\\color{red}{-4}-2)-5\\overset{?}{=}4(2(\\color{red}{-4})+1)+5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\"><\/td>\n<td style=\"height: 14px\">[latex]3(-6)-5\\overset{?}{=}4(-8+1)+5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\"><\/td>\n<td style=\"height: 14px\">[latex]-18-5\\overset{?}{=}4(-7)+5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\"><\/td>\n<td style=\"height: 14px\">[latex]-23\\overset{?}{=}-28+5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"height: 14px\"><\/td>\n<td style=\"height: 14px\">[latex]-23\\overset{?}{=}-23\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142586\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142586&#38;theme=oea&#38;iframe_resize_id=ohm142586&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm142587\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142587&#38;theme=oea&#38;iframe_resize_id=ohm142587&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve: [latex]\\Large\\frac{1}{2}\\normalsize\\left(6x - 2\\right)=5-x[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q735621\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q735621\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469851853\" class=\"unnumbered unstyled\" summary=\"The top line says one-half times parentheses 6x minus 2 equals 5 minus x. The next line says,\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{2}\\normalsize(6x-2)=5-x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Distribute.<\/td>\n<td style=\"height: 15px\">[latex]3x-1=5-x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Add [latex]x[\/latex] to get all the variables on the left.<\/td>\n<td style=\"height: 15px\">[latex]3x-1\\color{red}{+x}=5-x\\color{red}{+x}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Simplify.<\/td>\n<td style=\"height: 15px\">[latex]4x-1=5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Add [latex]1[\/latex] to get constants on the right.<\/td>\n<td style=\"height: 15px\">[latex]4x-1\\color{red}{+1}=5\\color{red}{+1}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Simplify.<\/td>\n<td style=\"height: 15px\">[latex]4x=6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Divide by [latex]4[\/latex].<\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{4x}{\\color{red}{4}}\\normalsize =\\Large\\frac{6}{\\color{red}{4}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Simplify.<\/td>\n<td style=\"height: 15px\">[latex]x=\\Large\\frac{3}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.8281px\">\n<td style=\"height: 15.8281px\">Check: Let [latex]x=\\Large\\frac{3}{2}[\/latex] .<\/td>\n<td style=\"height: 15.8281px\">\u00a0[latex]\\Large\\frac{1}{2}\\normalsize (6(\\Large\\frac{\\color{red}{3}}{\\color{red}{2}}\\normalsize )-2)\\overset{?}{=}5-(\\Large\\frac{\\color{red}3}{\\color{red}2})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{2}\\normalsize(9-2)\\overset{?}{=}\\Large\\frac{10}{2}\\normalsize -\\Large\\frac{3}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{2}\\normalsize(7)\\overset{?}{=}\\Large\\frac{7}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{7}{2}\\normalsize =\\Large\\frac{7}{2}\\normalsize\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142589\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142589&#38;theme=oea&#38;iframe_resize_id=ohm142589&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm142581\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142581&#38;theme=oea&#38;iframe_resize_id=ohm142581&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm142582\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142582&#38;theme=oea&#38;iframe_resize_id=ohm142582&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Watch the following video to see another example of how to solve an equation that requires distributing a fraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solve a Linear Equation with Parentheses and a Fraction 2\/3(9x-12)=8+2x\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1dmEoG7DkN4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next video example we show an example of solving an equation that requires distributing a fraction. \u00a0In this case, you will need to clear fractions after you distribute.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solve a Linear Equation with Parentheses and  Fractions 4\/5(2x+3)=5\/2x-3\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-P4KZECxo8Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q964378\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q964378\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468569426\" class=\"unnumbered unstyled\" summary=\"The top line says 0.24 times parentheses 100x plus 5 equals 0.4 times parentheses 30x plus 15. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]0.45a+0.36=0.3a+0.66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply by the least common denominator, 100<\/td>\n<td>[latex]45a+36=30a+66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]30a[\/latex] to get all the [latex]x[\/latex] s to the left.<\/td>\n<td>[latex]45a\\color{red}{-30a}+36=30a+66\\color{red}{-30a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]15a+36=66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]36[\/latex] to get the constants to the right.<\/td>\n<td>[latex]15a+36\\color{red}{-36}=66\\color{red}{-36}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]15a=30[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]\\large{\\frac{15a}{15}=\\frac{30}{15}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x = 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check: Let [latex]x=2[\/latex]<\/td>\n<td>\u00a0[latex]0.45\\left(2+0.8\\right)=0.3\\left(2+2.2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]1.26=1.26\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm140292\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=140292&theme=oea&iframe_resize_id=ohm140292&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video provides another example of how to solve an equation that requires distributing a decimal.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Solve a Linear Equation with Parentheses and Decimals 0.35(x-0.6)=0.2(x+1.2)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/k0K8mat_EaI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-4422\">\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>Solve a Linear Equation with Parentheses and Fractions 4\/5(2x+3)=5\/2x-3. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) fro Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-P4KZECxo8Y\">https:\/\/youtu.be\/-P4KZECxo8Y<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Solve a Linear Equation with Parentheses and a Fraction 2\/3(9x-12)=8+2x. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/1dmEoG7DkN4\">https:\/\/youtu.be\/1dmEoG7DkN4<\/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>Solve a Linear Equation with Parentheses and Decimals 0.35(x-0.6)=0.2(x+1.2). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/k0K8mat_EaI\">https:\/\/youtu.be\/k0K8mat_EaI<\/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>Question ID 142578, 142579, 142580, 142581, 142582, 142586, 142587, 142589. <strong>Authored 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>. <strong>License Terms<\/strong>:  IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1818. <strong>Authored by<\/strong>: Lawrence Morales. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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":25777,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Solve a Linear Equation with Parentheses and a Fraction 2\/3(9x-12)=8+2x\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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