{"id":365,"date":"2016-06-01T20:49:50","date_gmt":"2016-06-01T20:49:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=365"},"modified":"2018-01-31T03:23:02","modified_gmt":"2018-01-31T03:23:02","slug":"solving_linear_equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/solving_linear_equations\/","title":{"raw":"Solving Linear Equations","rendered":"Solving Linear Equations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use properties of equality to isolate variables and solve algebraic equations<\/li>\r\n \t<li>Use the properties of equality and the distributive property to solve equations\u00a0containing parentheses<\/li>\r\n \t<li>Clear fractions and decimals from equations to make them easier to solve<\/li>\r\n \t<li>Solve equations that have one solution, no solution, or an infinite number of solutions<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_4415\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-4415\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182803\/Screen-Shot-2016-05-27-at-1.13.50-PM-300x296.png\" alt=\"steps leading to a gold ball\" width=\"300\" height=\"296\" \/> Steps With an End In Sight[\/caption]\r\n<h2>Use properties of equality to isolate variables and solve algebraic equations<\/h2>\r\nThere are some equations that you can solve in your head quickly, but other equations are more complicated. Multi-step equations, ones that takes several steps to solve, can still be simplified and solved by applying basic algebraic rules such as the multiplication and addition properties of equality.\r\n\r\nIn this section we will explore methods for solving multi-step equations that contain grouping symbols and several mathematical operations. We will also learn techniques for solving multi-step equations that contain absolute values. Finally, we will\u00a0learn that\u00a0some equations have no solutions, while others have an infinite number of solutions.\r\n\r\nFirst, let's define some important terminology:\r\n<ul>\r\n \t<li><strong>variables:\u00a0<\/strong> variables are symbols that stand for an unknown quantity, they are often represented with letters, like <i>x<\/i>, <i>y<\/i>, or <i>z<\/i>.<\/li>\r\n \t<li><strong>coefficient:\u00a0<\/strong>Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3<i>x <\/i>is 3.<\/li>\r\n \t<li><strong>term:\u00a0<\/strong>a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[\/latex] are all terms<\/li>\r\n \t<li><strong>expression: <\/strong>groups of terms connected by addition and subtraction.\u00a0 [latex]2x^2-5[\/latex] is an expression<\/li>\r\n \t<li><strong>equation: <\/strong>\u00a0an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side.\u00a0Think of an equal sign as meaning \"the same as.\" Some examples of equations are\u00a0[latex]y = mx +b[\/latex], \u00a0[latex]\\frac{3}{4}r = v^{3} - r[\/latex], and \u00a0[latex]2(6-d) + f(3 +k) = \\frac{1}{4}d[\/latex]<\/li>\r\n<\/ul>\r\nThe following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[\/latex], the variable is [latex]x[\/latex], a coefficient is [latex]10[\/latex], a term is [latex]10x[\/latex], an expression is [latex]2x-3^2[\/latex].\r\n<div class=\"mceTemp\"><\/div>\r\n<img class=\"wp-image-4693 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/08214552\/Screen-Shot-2016-06-08-at-2.45.15-PM-300x242.png\" alt=\"Equation made of coefficients, variables, terms and expressions.\" width=\"424\" height=\"342\" \/>\r\n\r\nThere are some <b>equations<\/b> that you can solve in your head quickly. For example\u2014what is the value of <i>y<\/i> in the equation [latex]2y=6[\/latex]? Chances are you didn\u2019t need to get out a pencil and paper to calculate that [latex]y=3[\/latex]. You only needed to do one thing to get the answer: divide 6 by 2.\r\n\r\nOther equations are more complicated. Solving [latex]\\displaystyle 4\\left( \\frac{1}{3}t+\\frac{1}{2}\\right)=6[\/latex] without writing anything down is difficult! That\u2019s because this equation contains not just a <b>variable<\/b> but also fractions and <b>terms<\/b> inside parentheses. This is a <b>multi-step equation<\/b>, one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.\r\n\r\nRemember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The <b>addition property of equality<\/b> and the <b>multiplication property of equality<\/b> explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you\u2019ll keep both sides of the equation equal.\r\n\r\nIf the equation is in the form [latex]ax+b=c[\/latex], where <i>x<\/i> is the variable, you can solve the equation as before. First \u201cundo\u201d the addition and subtraction, and then \u201cundo\u201d the multiplication and division.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3y+2=11[\/latex].\r\n\r\n[reveal-answer q=\"843520\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843520\"]\r\n\r\nSubtract 2 from both sides of the equation to get the term with the variable by itself.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,\\,\\,\\,\\,-2}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/p>\r\nDivide both sides of the equation by 3 to get a coefficient of 1 for the variable.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3x+5x+4-x+7=88[\/latex].\r\n\r\n[reveal-answer q=\"455516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455516\"]\r\n\r\nThere are three like terms [latex]3x[\/latex], [latex]5x[\/latex],\u00a0and\u00a0[latex]\u2013x[\/latex]\u00a0involving a variable.\u00a0Combine these like terms.\u00a04 and 7 are also like terms and can be added.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+4+7=\\,\\,\\,88\\end{array}[\/latex]<\/p>\r\nThe equation is now in the form\u00a0[latex]ax+b=c[\/latex], so we can solve as before.\r\n<p style=\"text-align: center\">[latex]7x+11\\,\\,\\,=\\,\\,\\,88[\/latex]<\/p>\r\nSubtract 11 from both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\r\nDivide both sides by 7.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x=11[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nSome equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].\r\n\r\nTo solve this equation, we need to \u201cmove\u201d one of the variable terms.\u00a0This can make it difficult to decide which side to work with. It doesn\u2019t matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex], however, to avoid negative coefficients, you can move the smaller term.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nSolve:\u00a0[latex]4x-6=2x+10[\/latex]\r\n\r\n[reveal-answer q=\"457216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"457216\"]\r\n\r\nChoose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]\r\n<p style=\"text-align: center\">[latex]\\,\\,\\,4x-6=2x+10\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x}\\\\\\,\\,\\,2x-6=10[\/latex]<\/p>\r\n<p style=\"text-align: left\">Now add\u00a06 to both\u00a0sides to isolate the term with the variable.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\2x=16\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Now divide each side by 2 to isolate the variable x.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{2x}{2}=\\frac{16}{2}\\\\\\\\x=8\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this video, we show an example of solving equations that have variables on both sides of hte equal sign.\r\nhttps:\/\/youtu.be\/f3ujWNPL0Bw\r\n<h2>The Distributive Property<\/h2>\r\nAs we solve linear equations, we often need to do some work to write\u00a0the linear equations in a form we are familiar with solving.\u00a0This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.\r\n\r\nParentheses can\u00a0make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.\r\n<div class=\"textbox shaded\">\r\n<h3>The Distributive Property of Multiplication<\/h3>\r\nFor all real numbers <i>a, b,<\/i> and <i>c<\/i>,\u00a0[latex]a(b+c)=ab+ac[\/latex].\r\n\r\nWhat this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced\u00a0to <b>isolate the variable<\/b>\u00a0and solve the equation.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]a[\/latex]. [latex]4\\left(2a+3\\right)=28[\/latex]\r\n\r\n[reveal-answer q=\"372387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"372387\"]\r\n\r\nApply the distributive property to expand [latex]4\\left(2a+3\\right)[\/latex] to [latex]8a+12[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\r\nSubtract 12\u00a0from both sides to isolate\u00a0the variable term.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\r\nDivide both terms by 8 to get a coefficient of 1.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]a=2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.\r\n\r\nhttps:\/\/youtu.be\/aQOkD8L57V0\r\nIn the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]t[\/latex].\u00a0\u00a0[latex]2\\left(4t-5\\right)=-3\\left(2t+1\\right)[\/latex]\r\n\r\n[reveal-answer q=\"302387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"302387\"]\r\n\r\nApply the distributive property to expand [latex]2\\left(4t-5\\right)[\/latex] to [latex]8t-10[\/latex] and [latex]-3\\left(2t+1\\right)[\/latex] to[latex]-6t-3[\/latex]. Be careful in this step\u2014you are distributing a negative number, so keep track of the sign of each number after you multiply.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2\\left(4t-5\\right)=-3\\left(2t+1\\right)\\,\\,\\,\\,\\,\\, \\\\ 8t-10=-6t-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd [latex]-6t[\/latex] to both sides to begin combining like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides of the equation to isolate <em>t<\/em>.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t-10=-3\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 14t=\\,\\,\\,7\\,\\end{array}[\/latex]<\/p>\r\nThe last step is to divide both sides by 14 to completely isolate <em>t<\/em>.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}\\\\[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]t=\\frac{1}{2}\\\\[\/latex]\r\n\r\nWe simplified the fraction [latex]\\frac{7}{14}\\\\[\/latex] into [latex]\\frac{1}{2}\\\\[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we solve another multi-step equation with two sets of parentheses.\r\n\r\nhttps:\/\/youtu.be\/StomYTb7Xb8\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]72650[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation. This will clear all the fractions out of the equation. See the example below.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve \u00a0[latex]\\frac{1}{2}x-3=2-\\frac{3}{4}x[\/latex] by clearing the fractions in the equation first.\r\n\r\n[reveal-answer q=\"129951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"129951\"]\r\n\r\nMultiply both sides of the equation by 4, the common denominator of the fractional coefficients.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\frac{1}{2}x-3=2-\\frac{3}{4}x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\ 4\\left(\\frac{1}{2}x-3\\right)=4\\left(2-\\frac{3}{4}x\\right)\\end{array}[\/latex]<\/p>\r\nUse the distributive property to expand the expressions on both sides.\u00a0Multiply.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4\\left(\\frac{1}{2}x\\right)-4\\left(3\\right)=4\\left(2\\right)-4\\left(-\\frac{3}{4}x\\right)\\\\\\\\ \\frac{4}{2}x-12=8-\\frac{12}{4}x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\\\\\\\ 2x-12=8-3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\r\nAdd 3<em>x<\/em> to both sides to move the variable terms to only one side. Add 12 to both sides to move the variable\u00a0terms to only one side.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x-12=8-3x\\, \\\\\\underline{+3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+3x}\\\\ 5x-12=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 12 to both sides to move the <b>constant<\/b> terms to the other side.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5x-12=8\\,\\,\\\\ \\underline{\\,\\,\\,\\,\\,\\,+12\\,+12} \\\\5x=20\\end{array}[\/latex]<\/p>\r\nDivide to isolate the variable.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{5x}=\\underline{5}\\\\ 5\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\ x=4\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x=4[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nOf course, if you like to work with fractions, you can just apply your knowledge of operations with fractions and solve.\r\n\r\nIn the following video, we show how to solve a multi-step equation with fractions.\r\n\r\nhttps:\/\/youtu.be\/AvJTPeACTY0\r\n\r\nRegardless of which method you use to solve equations containing variables, you will get the same answer. You can choose the method you find the easiest! Remember to check your answer by substituting your solution into the original equation.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<p style=\"text-align: left\">[ohm_question]52524[\/ohm_question]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3y+10.5=6.5+2.5y[\/latex] by clearing the decimals in the equation first.\r\n\r\n[reveal-answer q=\"159951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"159951\"]\r\n\r\nSince the smallest decimal place represented in the equation is 0.10, we want to multiply by 10 to make 1.0\u00a0and clear\u00a0the decimals from the equation.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3y+10.5=6.5+2.5y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\ 10\\left(3y+10.5\\right)=10\\left(6.5+2.5y\\right)\\end{array}[\/latex]<\/p>\r\nUse the distributive property to expand the expressions on both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}10\\left(3y\\right)+10\\left(10.5\\right)=10\\left(6.5\\right)+10\\left(2.5y\\right)\\end{array}\\\\[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]30y+105=65+25y[\/latex]<\/p>\r\nMove the smaller variable term, [latex]25y[\/latex], by subtracting it from both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}30y+105=65+25y\\,\\,\\\\ \\underline{-25y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-25y} \\\\5y+105=65\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nSubtract 105 from both sides to isolate the term with the variable.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5y+105=65\\,\\,\\,\\\\ \\underline{\\,\\,\\,\\,\\,\\,-105\\,-105} \\\\5y=-40\\end{array}[\/latex]<\/p>\r\nDivide both sides by 5 to isolate the <em>y<\/em>.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\underline{5y}=\\underline{-40}\\\\ 5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\ \\,\\,\\,x=-8\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x=-8[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of clearing decimals first to solve a multi-step linear equation.\r\n\r\nhttps:\/\/youtu.be\/wtwepTZZnlY\r\nHere are some steps to follow when you solve multi-step equations.\r\n<div class=\"textbox shaded\">\r\n<h3>Solving Multi-Step Equations<\/h3>\r\n1. (Optional) Multiply to clear any fractions or decimals.\r\n\r\n2. Simplify each side by clearing parentheses and combining like terms.\r\n\r\n3. Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.\r\n\r\n4. Multiply or divide to isolate the variable.\r\n\r\n5. Check the solution.\r\n\r\n<\/div>\r\n<h2>Classify Solutions to Linear Equations<\/h2>\r\nThere are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that don't have any solutions, and even some that have an infinite number of solutions. The case where an equation has no\u00a0solution is illustrated in the next examples.\r\n<h2>Equations with no solutions<\/h2>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\u00a0[latex]12+2x\u20138=7x+5\u20135x[\/latex]\r\n\r\n[reveal-answer q=\"790409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790409\"]\r\n\r\nCombine <b>like terms<\/b> on both sides of the equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}12+2x-8=7x+5-5x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\end{array}[\/latex]<\/p>\r\nIsolate the <i>x<\/i> term by subtracting 2<i>x<\/i> from both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4= \\,5\\end{array}[\/latex]<\/p>\r\nThis false statement implies there are <strong>no solutions<\/strong> to this equation. Sometimes, we say the solution does not exist, or DNE for short.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis is <i>not<\/i> a solution! You did <i>not <\/i>find a value for <i>x<\/i>. Solving for <i>x<\/i> the way you know how, you arrive at the false statement [latex]4=5[\/latex]. Surely 4 cannot be equal to 5!\r\n\r\nThis may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by 2 and add 4 you would never get the same answer as when you multiply that same number by 2 and add 5. Since there is no value of <i>x <\/i>that will ever make this a true statement, the solution to the equation above is <i>\u201cno solution.\u201d<\/i>\r\n\r\nBe careful that you do not confuse the solution [latex]x=0[\/latex] with \u201cno solution.\u201d The solution [latex]x=0[\/latex]\u00a0means that the value 0 satisfies the equation, so there <i>is <\/i>a solution. \u201cNo solution\u201d means that there is no value, not even 0, which would satisfy the equation.\r\n\r\nAlso, be careful not to make the mistake of thinking that the equation [latex]4=5[\/latex] means that 4 and 5 are values for <i>x<\/i> that<i> <\/i>are solutions. If you substitute these values into the original equation, you\u2019ll see that they do not satisfy the equation. This is because there is truly <i>no solution<\/i>\u2014there are no values for <i>x<\/i> that will make the equation [latex]12+2x\u20138=7x+5\u20135x[\/latex]\u00a0true.\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Think\u00a0About It<\/h3>\r\nTry solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution?\r\n\r\na) Solve [latex]8y=3(y+4)+y[\/latex]\r\n\r\nUse the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"933839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933839\"]\r\n<p style=\"text-align: center\">Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\r\nFirst, distribute the 3 into the parentheses on the right-hand side.\r\n<p style=\"text-align: center\">[latex]8y=3(y+4)+y=8y=3y+12+y[\/latex]<\/p>\r\nNext, begin combining like terms.\r\n<p style=\"text-align: center\">[latex]8y=3y+12+y = 8y=4y+12[\/latex]<\/p>\r\nNow move the variable terms to one side. Moving the [latex]4y[\/latex] will help avoid a negative sign.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,8y=4y+12\\\\\\underline{-4y\\,\\,-4y}\\\\\\,\\,\\,\\,4y=12\\end{array}[\/latex]<\/p>\r\nNow, divide each side by [latex]4y[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{4y}{4}=\\frac{12}{4}\\\\y=3\\end{array}[\/latex]<\/p>\r\nBecause we were able to isolate <em>y<\/em> on one side and a number on the other side, we have one solution to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\nb) Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex]\r\n\r\nUse the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"937839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937839\"]\r\n\r\nSolve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex].\r\n\r\nFirst, distribute the 2 into the parentheses on the left-hand side.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2\\left(3x-5\\right)-4x=2x+7\\\\6x-10-4x=2x+7\\end{array}[\/latex]<\/p>\r\nNow begin simplifying. You can combine the <em>x<\/em> terms on the left-hand side.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}6x-10-4x=2x+7\\\\2x-10=2x+7\\end{array}[\/latex]<\/p>\r\nNow, take a moment to ponder this equation. It\u00a0says that [latex]2x-10[\/latex] is equal to [latex]2x+7[\/latex]. Can some number times two minus 10 be equal to that same number times two plus seven?\r\n\r\nLet's pretend [latex]x=3[\/latex].\r\n\r\nIs it true that\u00a0[latex]2\\left(3\\right)-10=-4[\/latex] is equal to\u00a0[latex]2\\left(3\\right)+7=13[\/latex]. NO! We don't even really need to continue solving the equation, but we can just to be thorough.\r\n\r\nAdd [latex]10[\/latex] to both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x-10=2x+7\\,\\,\\\\\\,\\,\\underline{+10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\2x=2x+17\\end{array}[\/latex]<\/p>\r\nNow move [latex]2x[\/latex] from the right hand side to combine like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,2x=2x+17\\\\\\,\\,\\underline{-2x\\,\\,-2x}\\\\\\,\\,\\,\\,\\,\\,\\,0=17\\end{array}[\/latex]<\/p>\r\nWe know that [latex]0\\text{ and }17[\/latex] are not equal, so there is no number that <em>x<\/em> could be to make this equation true.\r\n\r\nThis false statement implies there are <strong>no solutions<\/strong> to this equation, or DNE (does not exist) for short.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Algebraic Equations with an Infinite Number of Solutions<\/h3>\r\nYou have seen that if an equation has no solution, you end up with a false statement instead of a value for <i>x<\/i>. It is possible to have an equation where any value for <em>x<\/em> will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left\u00a0leaves us with an equation with exactly the same terms on both sides of the equal sign.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\u00a0[latex]5x+3\u20134x=3+x[\/latex]\r\n\r\n[reveal-answer q=\"773733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"773733\"]Combine like terms on both sides of the equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}5x+3-4x=3+x\\\\x+3=3+x\\end{array}[\/latex]<\/p>\r\nIsolate the <i>x<\/i> term by subtracting <i>x<\/i> from both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3=3+x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,=\\,\\,3\\end{array}[\/latex]<\/p>\r\nThis true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as \"All Real Numbers\"\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou arrive at the true statement \u201c[latex]3=3[\/latex].\u201d When you end up with a true statement like this, it means that the solution to the equation is \u201call real numbers.\u201d Try substituting [latex]x=0[\/latex]\u00a0into the original equation\u2014you will get a true statement! Try [latex]x=-\\frac{3}{4}[\/latex], and it also will check!\r\n\r\nThis equation happens to have an infinite number of solutions. Any value for <i>x <\/i>that you can think of will make this equation true. When you think about the context of the problem, this makes sense\u2014the equation [latex]x+3=3+x[\/latex] means \u201csome number plus 3 is equal to 3 plus that same number.\u201d We know that this is always true\u2014it\u2019s the commutative property of addition!\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\u00a0[latex]3\\left(2x-5\\right)=6x-15[\/latex]\r\n\r\n[reveal-answer q=\"973733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"973733\"]\r\n\r\nDistribute the 3 through the parentheses on the left-hand side.\r\n<p style=\"text-align: center\">[latex] \\begin{array}{r}3\\left(2x-5\\right)=6x-15\\\\6x-15=6x-15\\end{array}[\/latex]<\/p>\r\nWait! This looks just like the previous example. You have the same expression on both sides of an equal sign. \u00a0No matter what number you choose for <em>x<\/em>, you will have a true statement. We can finish the algebra:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6x-15=6x-15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-6x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-15\\,\\,=\\,\\,-15\\end{array}[\/latex]<\/p>\r\nThis true statement implies there are an infinite number of solutions to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more\u00a0examples of attempting to solve a linear equation with either no solution or many solutions.\r\n\r\nhttps:\/\/youtu.be\/iLkZ3o4wVxU\r\n\r\nIn the following video, we show more examples\u00a0of\u00a0solving linear equations with parentheses that have either no solution or many solutions.\r\n\r\nhttps:\/\/youtu.be\/EU_NEo1QBJ0\r\n<h2>Summary<\/h2>\r\nComplex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may need to simplify the equation first. This may mean using the distributive property to remove parentheses or multiplying both sides of an equation by a common denominator to get rid of fractions. Sometimes it requires both techniques. If your multi-step equation has an absolute value, you will need to solve two equations, sometimes isolating the absolute value expression first.\r\n\r\nWe have seen that solutions to equations can fall into three categories:\r\n<ul>\r\n \t<li>One solution<\/li>\r\n \t<li>No solution, DNE (does not exist)<\/li>\r\n \t<li>Many solutions, also called infinitely many solutions or All Real Numbers<\/li>\r\n<\/ul>\r\nAnd sometimes, we don't need to do much algebra to see what the outcome will be.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use properties of equality to isolate variables and solve algebraic equations<\/li>\n<li>Use the properties of equality and the distributive property to solve equations\u00a0containing parentheses<\/li>\n<li>Clear fractions and decimals from equations to make them easier to solve<\/li>\n<li>Solve equations that have one solution, no solution, or an infinite number of solutions<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_4415\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4415\" class=\"size-medium wp-image-4415\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182803\/Screen-Shot-2016-05-27-at-1.13.50-PM-300x296.png\" alt=\"steps leading to a gold ball\" width=\"300\" height=\"296\" \/><\/p>\n<p id=\"caption-attachment-4415\" class=\"wp-caption-text\">Steps With an End In Sight<\/p>\n<\/div>\n<h2>Use properties of equality to isolate variables and solve algebraic equations<\/h2>\n<p>There are some equations that you can solve in your head quickly, but other equations are more complicated. Multi-step equations, ones that takes several steps to solve, can still be simplified and solved by applying basic algebraic rules such as the multiplication and addition properties of equality.<\/p>\n<p>In this section we will explore methods for solving multi-step equations that contain grouping symbols and several mathematical operations. We will also learn techniques for solving multi-step equations that contain absolute values. Finally, we will\u00a0learn that\u00a0some equations have no solutions, while others have an infinite number of solutions.<\/p>\n<p>First, let&#8217;s define some important terminology:<\/p>\n<ul>\n<li><strong>variables:\u00a0<\/strong> variables are symbols that stand for an unknown quantity, they are often represented with letters, like <i>x<\/i>, <i>y<\/i>, or <i>z<\/i>.<\/li>\n<li><strong>coefficient:\u00a0<\/strong>Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3<i>x <\/i>is 3.<\/li>\n<li><strong>term:\u00a0<\/strong>a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[\/latex] are all terms<\/li>\n<li><strong>expression: <\/strong>groups of terms connected by addition and subtraction.\u00a0 [latex]2x^2-5[\/latex] is an expression<\/li>\n<li><strong>equation: <\/strong>\u00a0an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side.\u00a0Think of an equal sign as meaning &#8220;the same as.&#8221; Some examples of equations are\u00a0[latex]y = mx +b[\/latex], \u00a0[latex]\\frac{3}{4}r = v^{3} - r[\/latex], and \u00a0[latex]2(6-d) + f(3 +k) = \\frac{1}{4}d[\/latex]<\/li>\n<\/ul>\n<p>The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[\/latex], the variable is [latex]x[\/latex], a coefficient is [latex]10[\/latex], a term is [latex]10x[\/latex], an expression is [latex]2x-3^2[\/latex].<\/p>\n<div class=\"mceTemp\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4693 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/08214552\/Screen-Shot-2016-06-08-at-2.45.15-PM-300x242.png\" alt=\"Equation made of coefficients, variables, terms and expressions.\" width=\"424\" height=\"342\" \/><\/p>\n<p>There are some <b>equations<\/b> that you can solve in your head quickly. For example\u2014what is the value of <i>y<\/i> in the equation [latex]2y=6[\/latex]? Chances are you didn\u2019t need to get out a pencil and paper to calculate that [latex]y=3[\/latex]. You only needed to do one thing to get the answer: divide 6 by 2.<\/p>\n<p>Other equations are more complicated. Solving [latex]\\displaystyle 4\\left( \\frac{1}{3}t+\\frac{1}{2}\\right)=6[\/latex] without writing anything down is difficult! That\u2019s because this equation contains not just a <b>variable<\/b> but also fractions and <b>terms<\/b> inside parentheses. This is a <b>multi-step equation<\/b>, one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.<\/p>\n<p>Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The <b>addition property of equality<\/b> and the <b>multiplication property of equality<\/b> explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you\u2019ll keep both sides of the equation equal.<\/p>\n<p>If the equation is in the form [latex]ax+b=c[\/latex], where <i>x<\/i> is the variable, you can solve the equation as before. First \u201cundo\u201d the addition and subtraction, and then \u201cundo\u201d the multiplication and division.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]3y+2=11[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843520\">Show Solution<\/span><\/p>\n<div id=\"q843520\" class=\"hidden-answer\" style=\"display: none\">\n<p>Subtract 2 from both sides of the equation to get the term with the variable by itself.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,\\,\\,\\,\\,-2}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/p>\n<p>Divide both sides of the equation by 3 to get a coefficient of 1 for the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]3x+5x+4-x+7=88[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455516\">Show Solution<\/span><\/p>\n<div id=\"q455516\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are three like terms [latex]3x[\/latex], [latex]5x[\/latex],\u00a0and\u00a0[latex]\u2013x[\/latex]\u00a0involving a variable.\u00a0Combine these like terms.\u00a04 and 7 are also like terms and can be added.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+4+7=\\,\\,\\,88\\end{array}[\/latex]<\/p>\n<p>The equation is now in the form\u00a0[latex]ax+b=c[\/latex], so we can solve as before.<\/p>\n<p style=\"text-align: center\">[latex]7x+11\\,\\,\\,=\\,\\,\\,88[\/latex]<\/p>\n<p>Subtract 11 from both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\n<p>Divide both sides by 7.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=11[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].<\/p>\n<p>To solve this equation, we need to \u201cmove\u201d one of the variable terms.\u00a0This can make it difficult to decide which side to work with. It doesn\u2019t matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex], however, to avoid negative coefficients, you can move the smaller term.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Solve:\u00a0[latex]4x-6=2x+10[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q457216\">Show Solution<\/span><\/p>\n<div id=\"q457216\" class=\"hidden-answer\" style=\"display: none\">\n<p>Choose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\,\\,\\,4x-6=2x+10\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x}\\\\\\,\\,\\,2x-6=10[\/latex]<\/p>\n<p style=\"text-align: left\">Now add\u00a06 to both\u00a0sides to isolate the term with the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\2x=16\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Now divide each side by 2 to isolate the variable x.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{2x}{2}=\\frac{16}{2}\\\\\\\\x=8\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this video, we show an example of solving equations that have variables on both sides of hte equal sign.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solve an Equation with Variable on Both Sides\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/f3ujWNPL0Bw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Distributive Property<\/h2>\n<p>As we solve linear equations, we often need to do some work to write\u00a0the linear equations in a form we are familiar with solving.\u00a0This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.<\/p>\n<p>Parentheses can\u00a0make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Distributive Property of Multiplication<\/h3>\n<p>For all real numbers <i>a, b,<\/i> and <i>c<\/i>,\u00a0[latex]a(b+c)=ab+ac[\/latex].<\/p>\n<p>What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced\u00a0to <b>isolate the variable<\/b>\u00a0and solve the equation.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]a[\/latex]. [latex]4\\left(2a+3\\right)=28[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q372387\">Show Solution<\/span><\/p>\n<div id=\"q372387\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the distributive property to expand [latex]4\\left(2a+3\\right)[\/latex] to [latex]8a+12[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\n<p>Subtract 12\u00a0from both sides to isolate\u00a0the variable term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\n<p>Divide both terms by 8 to get a coefficient of 1.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]a=2[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solving an Equation with One Set of Parentheses\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aQOkD8L57V0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nIn the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]t[\/latex].\u00a0\u00a0[latex]2\\left(4t-5\\right)=-3\\left(2t+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q302387\">Show Solution<\/span><\/p>\n<div id=\"q302387\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the distributive property to expand [latex]2\\left(4t-5\\right)[\/latex] to [latex]8t-10[\/latex] and [latex]-3\\left(2t+1\\right)[\/latex] to[latex]-6t-3[\/latex]. Be careful in this step\u2014you are distributing a negative number, so keep track of the sign of each number after you multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2\\left(4t-5\\right)=-3\\left(2t+1\\right)\\,\\,\\,\\,\\,\\, \\\\ 8t-10=-6t-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add [latex]-6t[\/latex] to both sides to begin combining like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add 10 to both sides of the equation to isolate <em>t<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t-10=-3\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 14t=\\,\\,\\,7\\,\\end{array}[\/latex]<\/p>\n<p>The last step is to divide both sides by 14 to completely isolate <em>t<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}\\\\[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]t=\\frac{1}{2}\\\\[\/latex]<\/p>\n<p>We simplified the fraction [latex]\\frac{7}{14}\\\\[\/latex] into [latex]\\frac{1}{2}\\\\[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the following video, we solve another multi-step equation with two sets of parentheses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Solving an Equation with Parentheses on Both Sides\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/StomYTb7Xb8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm72650\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=72650&theme=oea&iframe_resize_id=ohm72650&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation. This will clear all the fractions out of the equation. See the example below.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve \u00a0[latex]\\frac{1}{2}x-3=2-\\frac{3}{4}x[\/latex] by clearing the fractions in the equation first.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q129951\">Show Solution<\/span><\/p>\n<div id=\"q129951\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply both sides of the equation by 4, the common denominator of the fractional coefficients.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\frac{1}{2}x-3=2-\\frac{3}{4}x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\ 4\\left(\\frac{1}{2}x-3\\right)=4\\left(2-\\frac{3}{4}x\\right)\\end{array}[\/latex]<\/p>\n<p>Use the distributive property to expand the expressions on both sides.\u00a0Multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4\\left(\\frac{1}{2}x\\right)-4\\left(3\\right)=4\\left(2\\right)-4\\left(-\\frac{3}{4}x\\right)\\\\\\\\ \\frac{4}{2}x-12=8-\\frac{12}{4}x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\\\\\\\ 2x-12=8-3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\n<p>Add 3<em>x<\/em> to both sides to move the variable terms to only one side. Add 12 to both sides to move the variable\u00a0terms to only one side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x-12=8-3x\\, \\\\\\underline{+3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+3x}\\\\ 5x-12=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add 12 to both sides to move the <b>constant<\/b> terms to the other side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5x-12=8\\,\\,\\\\ \\underline{\\,\\,\\,\\,\\,\\,+12\\,+12} \\\\5x=20\\end{array}[\/latex]<\/p>\n<p>Divide to isolate the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{5x}=\\underline{5}\\\\ 5\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\ x=4\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=4[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Of course, if you like to work with fractions, you can just apply your knowledge of operations with fractions and solve.<\/p>\n<p>In the following video, we show how to solve a multi-step equation with fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Solving an Equation with Fractions (Clear Fractions)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/AvJTPeACTY0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Regardless of which method you use to solve equations containing variables, you will get the same answer. You can choose the method you find the easiest! Remember to check your answer by substituting your solution into the original equation.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p style=\"text-align: left\"><iframe loading=\"lazy\" id=\"ohm52524\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=52524&theme=oea&iframe_resize_id=ohm52524&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Sometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]3y+10.5=6.5+2.5y[\/latex] by clearing the decimals in the equation first.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q159951\">Show Solution<\/span><\/p>\n<div id=\"q159951\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the smallest decimal place represented in the equation is 0.10, we want to multiply by 10 to make 1.0\u00a0and clear\u00a0the decimals from the equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3y+10.5=6.5+2.5y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\ 10\\left(3y+10.5\\right)=10\\left(6.5+2.5y\\right)\\end{array}[\/latex]<\/p>\n<p>Use the distributive property to expand the expressions on both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}10\\left(3y\\right)+10\\left(10.5\\right)=10\\left(6.5\\right)+10\\left(2.5y\\right)\\end{array}\\\\[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]30y+105=65+25y[\/latex]<\/p>\n<p>Move the smaller variable term, [latex]25y[\/latex], by subtracting it from both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}30y+105=65+25y\\,\\,\\\\ \\underline{-25y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-25y} \\\\5y+105=65\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Subtract 105 from both sides to isolate the term with the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5y+105=65\\,\\,\\,\\\\ \\underline{\\,\\,\\,\\,\\,\\,-105\\,-105} \\\\5y=-40\\end{array}[\/latex]<\/p>\n<p>Divide both sides by 5 to isolate the <em>y<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\underline{5y}=\\underline{-40}\\\\ 5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\ \\,\\,\\,x=-8\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=-8[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of clearing decimals first to solve a multi-step linear equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Solving an Equation with Decimals (Clear Decimals)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/wtwepTZZnlY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nHere are some steps to follow when you solve multi-step equations.<\/p>\n<div class=\"textbox shaded\">\n<h3>Solving Multi-Step Equations<\/h3>\n<p>1. (Optional) Multiply to clear any fractions or decimals.<\/p>\n<p>2. Simplify each side by clearing parentheses and combining like terms.<\/p>\n<p>3. Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.<\/p>\n<p>4. Multiply or divide to isolate the variable.<\/p>\n<p>5. Check the solution.<\/p>\n<\/div>\n<h2>Classify Solutions to Linear Equations<\/h2>\n<p>There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that don&#8217;t have any solutions, and even some that have an infinite number of solutions. The case where an equation has no\u00a0solution is illustrated in the next examples.<\/p>\n<h2>Equations with no solutions<\/h2>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.\u00a0[latex]12+2x\u20138=7x+5\u20135x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q790409\">Show Solution<\/span><\/p>\n<div id=\"q790409\" class=\"hidden-answer\" style=\"display: none\">\n<p>Combine <b>like terms<\/b> on both sides of the equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}12+2x-8=7x+5-5x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\end{array}[\/latex]<\/p>\n<p>Isolate the <i>x<\/i> term by subtracting 2<i>x<\/i> from both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4= \\,5\\end{array}[\/latex]<\/p>\n<p>This false statement implies there are <strong>no solutions<\/strong> to this equation. Sometimes, we say the solution does not exist, or DNE for short.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This is <i>not<\/i> a solution! You did <i>not <\/i>find a value for <i>x<\/i>. Solving for <i>x<\/i> the way you know how, you arrive at the false statement [latex]4=5[\/latex]. Surely 4 cannot be equal to 5!<\/p>\n<p>This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by 2 and add 4 you would never get the same answer as when you multiply that same number by 2 and add 5. Since there is no value of <i>x <\/i>that will ever make this a true statement, the solution to the equation above is <i>\u201cno solution.\u201d<\/i><\/p>\n<p>Be careful that you do not confuse the solution [latex]x=0[\/latex] with \u201cno solution.\u201d The solution [latex]x=0[\/latex]\u00a0means that the value 0 satisfies the equation, so there <i>is <\/i>a solution. \u201cNo solution\u201d means that there is no value, not even 0, which would satisfy the equation.<\/p>\n<p>Also, be careful not to make the mistake of thinking that the equation [latex]4=5[\/latex] means that 4 and 5 are values for <i>x<\/i> that<i> <\/i>are solutions. If you substitute these values into the original equation, you\u2019ll see that they do not satisfy the equation. This is because there is truly <i>no solution<\/i>\u2014there are no values for <i>x<\/i> that will make the equation [latex]12+2x\u20138=7x+5\u20135x[\/latex]\u00a0true.<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Think\u00a0About It<\/h3>\n<p>Try solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution?<\/p>\n<p>a) Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\n<p>Use the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933839\">Show Solution<\/span><\/p>\n<div id=\"q933839\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\n<p>First, distribute the 3 into the parentheses on the right-hand side.<\/p>\n<p style=\"text-align: center\">[latex]8y=3(y+4)+y=8y=3y+12+y[\/latex]<\/p>\n<p>Next, begin combining like terms.<\/p>\n<p style=\"text-align: center\">[latex]8y=3y+12+y = 8y=4y+12[\/latex]<\/p>\n<p>Now move the variable terms to one side. Moving the [latex]4y[\/latex] will help avoid a negative sign.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,8y=4y+12\\\\\\underline{-4y\\,\\,-4y}\\\\\\,\\,\\,\\,4y=12\\end{array}[\/latex]<\/p>\n<p>Now, divide each side by [latex]4y[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{4y}{4}=\\frac{12}{4}\\\\y=3\\end{array}[\/latex]<\/p>\n<p>Because we were able to isolate <em>y<\/em> on one side and a number on the other side, we have one solution to this equation.<\/p>\n<\/div>\n<\/div>\n<p>b) Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex]<\/p>\n<p>Use the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q937839\">Show Solution<\/span><\/p>\n<div id=\"q937839\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex].<\/p>\n<p>First, distribute the 2 into the parentheses on the left-hand side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2\\left(3x-5\\right)-4x=2x+7\\\\6x-10-4x=2x+7\\end{array}[\/latex]<\/p>\n<p>Now begin simplifying. You can combine the <em>x<\/em> terms on the left-hand side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}6x-10-4x=2x+7\\\\2x-10=2x+7\\end{array}[\/latex]<\/p>\n<p>Now, take a moment to ponder this equation. It\u00a0says that [latex]2x-10[\/latex] is equal to [latex]2x+7[\/latex]. Can some number times two minus 10 be equal to that same number times two plus seven?<\/p>\n<p>Let&#8217;s pretend [latex]x=3[\/latex].<\/p>\n<p>Is it true that\u00a0[latex]2\\left(3\\right)-10=-4[\/latex] is equal to\u00a0[latex]2\\left(3\\right)+7=13[\/latex]. NO! We don&#8217;t even really need to continue solving the equation, but we can just to be thorough.<\/p>\n<p>Add [latex]10[\/latex] to both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x-10=2x+7\\,\\,\\\\\\,\\,\\underline{+10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\2x=2x+17\\end{array}[\/latex]<\/p>\n<p>Now move [latex]2x[\/latex] from the right hand side to combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,2x=2x+17\\\\\\,\\,\\underline{-2x\\,\\,-2x}\\\\\\,\\,\\,\\,\\,\\,\\,0=17\\end{array}[\/latex]<\/p>\n<p>We know that [latex]0\\text{ and }17[\/latex] are not equal, so there is no number that <em>x<\/em> could be to make this equation true.<\/p>\n<p>This false statement implies there are <strong>no solutions<\/strong> to this equation, or DNE (does not exist) for short.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Algebraic Equations with an Infinite Number of Solutions<\/h3>\n<p>You have seen that if an equation has no solution, you end up with a false statement instead of a value for <i>x<\/i>. It is possible to have an equation where any value for <em>x<\/em> will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left\u00a0leaves us with an equation with exactly the same terms on both sides of the equal sign.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.\u00a0[latex]5x+3\u20134x=3+x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q773733\">Show Solution<\/span><\/p>\n<div id=\"q773733\" class=\"hidden-answer\" style=\"display: none\">Combine like terms on both sides of the equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}5x+3-4x=3+x\\\\x+3=3+x\\end{array}[\/latex]<\/p>\n<p>Isolate the <i>x<\/i> term by subtracting <i>x<\/i> from both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3=3+x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,=\\,\\,3\\end{array}[\/latex]<\/p>\n<p>This true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as &#8220;All Real Numbers&#8221;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You arrive at the true statement \u201c[latex]3=3[\/latex].\u201d When you end up with a true statement like this, it means that the solution to the equation is \u201call real numbers.\u201d Try substituting [latex]x=0[\/latex]\u00a0into the original equation\u2014you will get a true statement! Try [latex]x=-\\frac{3}{4}[\/latex], and it also will check!<\/p>\n<p>This equation happens to have an infinite number of solutions. Any value for <i>x <\/i>that you can think of will make this equation true. When you think about the context of the problem, this makes sense\u2014the equation [latex]x+3=3+x[\/latex] means \u201csome number plus 3 is equal to 3 plus that same number.\u201d We know that this is always true\u2014it\u2019s the commutative property of addition!<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.\u00a0[latex]3\\left(2x-5\\right)=6x-15[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q973733\">Show Solution<\/span><\/p>\n<div id=\"q973733\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute the 3 through the parentheses on the left-hand side.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3\\left(2x-5\\right)=6x-15\\\\6x-15=6x-15\\end{array}[\/latex]<\/p>\n<p>Wait! This looks just like the previous example. You have the same expression on both sides of an equal sign. \u00a0No matter what number you choose for <em>x<\/em>, you will have a true statement. We can finish the algebra:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6x-15=6x-15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-6x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-15\\,\\,=\\,\\,-15\\end{array}[\/latex]<\/p>\n<p>This true statement implies there are an infinite number of solutions to this equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more\u00a0examples of attempting to solve a linear equation with either no solution or many solutions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Linear Equations with No Solutions or Infinite Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iLkZ3o4wVxU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video, we show more examples\u00a0of\u00a0solving linear equations with parentheses that have either no solution or many solutions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Linear Equations with No Solutions of Infinite Solutions (Parentheses)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EU_NEo1QBJ0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Complex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may need to simplify the equation first. This may mean using the distributive property to remove parentheses or multiplying both sides of an equation by a common denominator to get rid of fractions. Sometimes it requires both techniques. If your multi-step equation has an absolute value, you will need to solve two equations, sometimes isolating the absolute value expression first.<\/p>\n<p>We have seen that solutions to equations can fall into three categories:<\/p>\n<ul>\n<li>One solution<\/li>\n<li>No solution, DNE (does not exist)<\/li>\n<li>Many solutions, also called infinitely many solutions or All Real Numbers<\/li>\n<\/ul>\n<p>And sometimes, we don&#8217;t need to do much algebra to see what the outcome will be.<\/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-365\">\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><strong>Provided by<\/strong>: LumenLearning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Steps With an End In Sight. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving Two Step Equations (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/fCyxSVQKeRw\">https:\/\/youtu.be\/fCyxSVQKeRw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation that Requires Combining Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ez_sP2OTGjU\">https:\/\/youtu.be\/ez_sP2OTGjU<\/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 an Equation with Variable on Both Sides. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/f3ujWNPL0Bw\">https:\/\/youtu.be\/f3ujWNPL0Bw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with One Set of Parentheses. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/aQOkD8L57V0\">https:\/\/youtu.be\/aQOkD8L57V0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with Parentheses on Both Sides. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/StomYTb7Xb8\">https:\/\/youtu.be\/StomYTb7Xb8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with Fractions (Clear Fractions). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/AvJTPeACTY0\">https:\/\/youtu.be\/AvJTPeACTY0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with Decimals (Clear Decimals). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/wtwepTZZnlY\">https:\/\/youtu.be\/wtwepTZZnlY<\/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>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 4: Solving Absolute Value Equations (Requires Isolating Abs. Value). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-HrOMkIiSfU\">https:\/\/youtu.be\/-HrOMkIiSfU<\/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 72650. <strong>Authored by<\/strong>: Hidegkuti,Marta. <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#52524. <strong>Authored by<\/strong>: Hidegkuti,Marta. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"LumenLearning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Steps With an End In Sight\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solving Two Step Equations (Basic)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/fCyxSVQKeRw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solving an Equation that Requires Combining Like Terms\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ez_sP2OTGjU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solve an Equation with Variable on Both Sides\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/f3ujWNPL0Bw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 4: Solving Absolute Value Equations (Requires Isolating Abs. Value)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/-HrOMkIiSfU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solving an Equation with One Set of Parentheses\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/aQOkD8L57V0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solving an Equation with Parentheses on Both Sides\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/StomYTb7Xb8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solving an Equation with Fractions (Clear Fractions)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/AvJTPeACTY0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solving an Equation with Decimals (Clear Decimals)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/wtwepTZZnlY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 72650\",\"author\":\"Hidegkuti,Marta\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID#52524\",\"author\":\"Hidegkuti,Marta\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"834ccc98-e998-4489-90d5-c315603e0de9","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-365","chapter","type-chapter","status-publish","hentry"],"part":359,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/365","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/365\/revisions"}],"predecessor-version":[{"id":5190,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/365\/revisions\/5190"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/359"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/365\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=365"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=365"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=365"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}