{"id":2850,"date":"2024-02-09T19:36:51","date_gmt":"2024-02-09T19:36:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2850"},"modified":"2026-03-09T16:54:03","modified_gmt":"2026-03-09T16:54:03","slug":"6-1-3-a-general-strategy-to-solve-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-1-3-a-general-strategy-to-solve-linear-equations\/","title":{"raw":"6.1.3 A General Strategy to Solve Linear Equations","rendered":"6.1.3 A General Strategy to Solve Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Solve a linear equation that requires multiple steps and a combination of the properties of equality<\/li>\r\n \t<li>Solve equations with fraction coefficients<\/li>\r\n \t<li>Solve equations with decimal coefficients<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key WORDS<\/h1>\r\n<ul>\r\n \t<li><strong>General Strategy<\/strong>: a plan that can be followed that works in all cases<\/li>\r\n<\/ul>\r\n<\/div>\r\nIt\u2019s time now to lay out an overall strategy that can be used to solve <em>any<\/em> linear equation in one variable. We call this the <em><strong>general strategy<\/strong><\/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 in one variable<\/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 equal to [latex]1[\/latex]. Use the Multiplication or Division Property of Equality.<\/li>\r\n \t<li>State the solution to the equation. If there is a contradiction, there is no solution. If there is an identity, the solution is the set of all real numbers.<\/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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]3\\left(x+2\\right)=18[\/latex]<\/th>\r\n<\/tr>\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]\\frac{3x}{\\color{red}{3}}\\normalsize =\\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<h4>Answer<\/h4>\r\n[latex]x=4[\/latex]\r\n\r\n<\/div>\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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]-\\left(x+5\\right)=7[\/latex]<\/th>\r\n<\/tr>\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<strong>Answer<\/strong>\r\n\r\n[latex]x=-12[\/latex]\r\n\r\n<\/div>\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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]4\\left(x - 2\\right)+5=-3[\/latex]<\/th>\r\n<\/tr>\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]\\frac{4x}{\\color{red}{4}} =\\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<h4>Answer<\/h4>\r\n[latex]x=0[\/latex]\r\n\r\n<\/div>\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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]8 - 2\\left(3y+5\\right)=0[\/latex]<\/th>\r\n<\/tr>\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]\\\\frac{-6y}{\\color{red}{-6}} =\\frac{2}{\\color{red}{-6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=-\\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}{-\\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<h4>Answer<\/h4>\r\n[latex]y=-\\frac{1}{3}[\/latex]\r\n\r\n<\/div>\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<h4>Solution<\/h4>\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>\r\n<th colspan=\"2\">Solving [latex]3\\left(x - 2\\right)-5=4\\left(2x+1\\right)+5[\/latex]<\/th>\r\n<\/tr>\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]\\frac{-20}{\\color{red}{5}} =\\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<h4>Answer<\/h4>\r\n[latex]x=-4[\/latex]\r\n\r\n<\/div>\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]\\frac{1}{2}\\left(6x - 2\\right)=5-x[\/latex]\r\n<h4>Solution<\/h4>\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>\r\n<th colspan=\"2\">Solving [latex]\\frac{1}{2}\\left(6x - 2\\right)=5-x[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{1}{2}(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]\\frac{4x}{\\color{red}{4}} =\\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=\\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=\\frac{3}{2}[\/latex] .<\/td>\r\n<td style=\"height: 15.8281px;\">\u00a0[latex]\\frac{1}{2} (6(\\frac{\\color{red}{3}}{\\color{red}{2}} )-2)\\overset{?}{=}5-(\\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]\\frac{1}{2}(9-2)\\overset{?}{=}\\frac{10}{2} -\\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]\\frac{1}{2}(7)\\overset{?}{=\\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;\">[latexe\\frac{7}{2} =\\frac{7}{2}\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\n[latex]x=\\frac{3}{2}[\/latex]\r\n\r\n<\/div>\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\n&nbsp;\r\n\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\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.3-1.docx\">Transcript-6.1.3-1<\/a>\r\n\r\nIn many applications, we will have to solve equations with decimals. The same general strategy will work for these equations. We can choose to work with the decimals, or to clear the decimals. To clear decimals, we multiply both sides of the equation by an appropriate power of 10. The power is determined by the decimal with the most decimal places after the point.\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<h4>Solution<\/h4>\r\nSince the longest decimal has 2 decimal places after the point, we multiply by 100.\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<th style=\"width: 410.2px;\" colspan=\"2\">Solving [latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\"><\/td>\r\n<td style=\"width: 258.1px;\">[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 style=\"width: 140.938px;\">Distribute.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]0.45a+0.36=0.3a+0.66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Multiply by the least common denominator, 100<\/td>\r\n<td style=\"width: 258.1px;\">[latex]45a+36=30a+66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Subtract [latex]30a[\/latex] to get all the [latex]x[\/latex] s to the left.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]45a\\color{red}{-30a}+36=30a+66\\color{red}{-30a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Simplify.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]15a+36=66[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Subtract [latex]36[\/latex] to get the constants to the right.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]15a+36\\color{red}{-36}=66\\color{red}{-36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Simplify.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]15a=30[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Divide.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]\\large{\\frac{15a}{15}=\\frac{30}{15}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Simplify.<\/td>\r\n<td style=\"width: 258.1px;\">[latex]a = 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 140.938px;\">Check: Let [latex]a=2[\/latex]<\/td>\r\n<td style=\"width: 258.1px;\">\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 style=\"width: 140.938px;\"><\/td>\r\n<td style=\"width: 258.1px;\">\u00a0[latex]1.26=1.26\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\n[latex]a = 2[\/latex]\r\n\r\n<\/div>\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\n&nbsp;\r\n\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\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.3-2.docx\">Transcript-6.1.3-2<\/a>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nSolve:\r\n<ol>\r\n \t<li>[latex]5(4-x)=-3(x+6)-2x[\/latex]<\/li>\r\n \t<li>[latex]3-(5x-7)=2(5-7x)+9x[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n1.\r\n\r\n[latex]\\begin{equation}\\begin{aligned}5(4-x)&amp;=-3(x+6)-2x \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Distribute} \\\\20-5x&amp;=-3x-18-2x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Combine like terms}\\\\20-5x&amp;=-5x-18\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Add 5x to both sides}\\\\20-5x+5x&amp;=-18\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Combine like terms}\\\\ 20&amp;=-18\\end{aligned}\\end{equation}[\/latex]\r\n\r\nContradiction. No solution.\r\n\r\n&nbsp;\r\n\r\n2.\r\n\r\n[latex]\\begin{equation}\\begin{aligned}3-(5x-7)&amp;=2(5-7x)+9x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Distribute}\\\\ 3-5x+7&amp;=10-14x+9x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Combine like terms}\\\\ 10-5x&amp;=10-5x\\end{aligned}\\end{equation}[\/latex]\r\n\r\nIdentity. Solution is all real numbers.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve:\r\n<ol>\r\n \t<li>[latex]3(x+7)=2(x+6)+x[\/latex]<\/li>\r\n \t<li>[latex]9x-(3x-7)=2(3x-4)+15[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm481\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm481\"]\r\n<ol>\r\n \t<li>Contradiction. No solution.<\/li>\r\n \t<li>Identity. Solution is all real numbers.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Solve a linear equation that requires multiple steps and a combination of the properties of equality<\/li>\n<li>Solve equations with fraction coefficients<\/li>\n<li>Solve equations with decimal coefficients<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key WORDS<\/h1>\n<ul>\n<li><strong>General Strategy<\/strong>: a plan that can be followed that works in all cases<\/li>\n<\/ul>\n<\/div>\n<p>It\u2019s time now to lay out an overall strategy that can be used to solve <em>any<\/em> linear equation in one variable. We call this the <em><strong>general strategy<\/strong><\/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 in one variable<\/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 equal to [latex]1[\/latex]. Use the Multiplication or Division Property of Equality.<\/li>\n<li>State the solution to the equation. If there is a contradiction, there is no solution. If there is an identity, the solution is the set of all real numbers.<\/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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]3\\left(x+2\\right)=18[\/latex]<\/th>\n<\/tr>\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]\\frac{3x}{\\color{red}{3}}\\normalsize =\\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<h4>Answer<\/h4>\n<p>[latex]x=4[\/latex]<\/p>\n<\/div>\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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]-\\left(x+5\\right)=7[\/latex]<\/th>\n<\/tr>\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<p><strong>Answer<\/strong><\/p>\n<p>[latex]x=-12[\/latex]<\/p>\n<\/div>\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<h4>Solution<\/h4>\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<th colspan=\"2\">Solving [latex]4\\left(x - 2\\right)+5=-3[\/latex]<\/th>\n<\/tr>\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]\\frac{4x}{\\color{red}{4}} =\\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<h4>Answer<\/h4>\n<p>[latex]x=0[\/latex]<\/p>\n<\/div>\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<h4>Solution<\/h4>\n<p>Be 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<th colspan=\"2\">Solving [latex]8 - 2\\left(3y+5\\right)=0[\/latex]<\/th>\n<\/tr>\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]\\\\frac{-6y}{\\color{red}{-6}} =\\frac{2}{\\color{red}{-6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=-\\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}{-\\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<h4>Answer<\/h4>\n<p>[latex]y=-\\frac{1}{3}[\/latex]<\/p>\n<\/div>\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<h4>Solution<\/h4>\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>\n<th colspan=\"2\">Solving [latex]3\\left(x - 2\\right)-5=4\\left(2x+1\\right)+5[\/latex]<\/th>\n<\/tr>\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]\\frac{-20}{\\color{red}{5}} =\\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<h4>Answer<\/h4>\n<p>[latex]x=-4[\/latex]<\/p>\n<\/div>\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]\\frac{1}{2}\\left(6x - 2\\right)=5-x[\/latex]<\/p>\n<h4>Solution<\/h4>\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>\n<th colspan=\"2\">Solving [latex]\\frac{1}{2}\\left(6x - 2\\right)=5-x[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]\\frac{1}{2}(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]\\frac{4x}{\\color{red}{4}} =\\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=\\frac{3}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.8281px;\">\n<td style=\"height: 15.8281px;\">Check: Let [latex]x=\\frac{3}{2}[\/latex] .<\/td>\n<td style=\"height: 15.8281px;\">\u00a0[latex]\\frac{1}{2} (6(\\frac{\\color{red}{3}}{\\color{red}{2}} )-2)\\overset{?}{=}5-(\\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]\\frac{1}{2}(9-2)\\overset{?}{=}\\frac{10}{2} -\\frac{3}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]\\frac{1}{2}(7)\\overset{?}{=\\frac{7}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latexe\\frac{7}{2} =\\frac{7}{2}\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>[latex]x=\\frac{3}{2}[\/latex]<\/p>\n<\/div>\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>&nbsp;<\/p>\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><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.3-1.docx\">Transcript-6.1.3-1<\/a><\/p>\n<p>In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations. We can choose to work with the decimals, or to clear the decimals. To clear decimals, we multiply both sides of the equation by an appropriate power of 10. The power is determined by the decimal with the most decimal places after the point.<\/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<h4>Solution<\/h4>\n<p>Since the longest decimal has 2 decimal places after the point, we multiply by 100.<\/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<th style=\"width: 410.2px;\" colspan=\"2\">Solving [latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\"><\/td>\n<td style=\"width: 258.1px;\">[latex]0.45\\left(a+0.8\\right)=0.3\\left(a+2.2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Distribute.<\/td>\n<td style=\"width: 258.1px;\">[latex]0.45a+0.36=0.3a+0.66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Multiply by the least common denominator, 100<\/td>\n<td style=\"width: 258.1px;\">[latex]45a+36=30a+66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Subtract [latex]30a[\/latex] to get all the [latex]x[\/latex] s to the left.<\/td>\n<td style=\"width: 258.1px;\">[latex]45a\\color{red}{-30a}+36=30a+66\\color{red}{-30a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Simplify.<\/td>\n<td style=\"width: 258.1px;\">[latex]15a+36=66[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Subtract [latex]36[\/latex] to get the constants to the right.<\/td>\n<td style=\"width: 258.1px;\">[latex]15a+36\\color{red}{-36}=66\\color{red}{-36}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Simplify.<\/td>\n<td style=\"width: 258.1px;\">[latex]15a=30[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Divide.<\/td>\n<td style=\"width: 258.1px;\">[latex]\\large{\\frac{15a}{15}=\\frac{30}{15}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Simplify.<\/td>\n<td style=\"width: 258.1px;\">[latex]a = 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\">Check: Let [latex]a=2[\/latex]<\/td>\n<td style=\"width: 258.1px;\">\u00a0[latex]0.45\\left(2+0.8\\right)=0.3\\left(2+2.2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 140.938px;\"><\/td>\n<td style=\"width: 258.1px;\">\u00a0[latex]1.26=1.26\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>[latex]a = 2[\/latex]<\/p>\n<\/div>\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>&nbsp;<\/p>\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-2\" 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<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.3-2.docx\">Transcript-6.1.3-2<\/a><\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Solve:<\/p>\n<ol>\n<li>[latex]5(4-x)=-3(x+6)-2x[\/latex]<\/li>\n<li>[latex]3-(5x-7)=2(5-7x)+9x[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>1.<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}5(4-x)&=-3(x+6)-2x \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Distribute} \\\\20-5x&=-3x-18-2x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Combine like terms}\\\\20-5x&=-5x-18\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Add 5x to both sides}\\\\20-5x+5x&=-18\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Combine like terms}\\\\ 20&=-18\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>Contradiction. No solution.<\/p>\n<p>&nbsp;<\/p>\n<p>2.<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}3-(5x-7)&=2(5-7x)+9x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Distribute}\\\\ 3-5x+7&=10-14x+9x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{Combine like terms}\\\\ 10-5x&=10-5x\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>Identity. Solution is all real numbers.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve:<\/p>\n<ol>\n<li>[latex]3(x+7)=2(x+6)+x[\/latex]<\/li>\n<li>[latex]9x-(3x-7)=2(3x-4)+15[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm481\">Show Answer<\/span><\/p>\n<div id=\"qhjm481\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Contradiction. No solution.<\/li>\n<li>Identity. Solution is all real numbers.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":370291,"menu_order":3,"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-2850","chapter","type-chapter","status-publish","hentry"],"part":2842,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2850","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2850\/revisions"}],"predecessor-version":[{"id":3069,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2850\/revisions\/3069"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2842"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2850\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2850"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2850"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2850"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}