{"id":16110,"date":"2019-10-01T17:41:07","date_gmt":"2019-10-01T17:41:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/using-the-addition-property-of-equality\/"},"modified":"2024-04-30T21:26:50","modified_gmt":"2024-04-30T21:26:50","slug":"using-the-addition-property-of-equality","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/using-the-addition-property-of-equality\/","title":{"raw":"Classifying Solutions to Linear Equations","rendered":"Classifying Solutions to Linear Equations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\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\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 [latex]x.[\/latex]\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 [latex]x[\/latex] term by subtracting [latex]2x[\/latex] 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, because [latex]4\\not=5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThis is <i>not<\/i> a solution! You did <i>not<\/i> find a value for\u00a0[latex]x[\/latex]. Solving for [latex]x[\/latex] the way you know how, you arrive at the false statement [latex]4=5[\/latex]. Surely \u00a0[latex]4[\/latex] cannot be equal to [latex]5[\/latex]!\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 [latex]2[\/latex] and add\u00a0[latex]4[\/latex], you would never get the same answer as when you multiply that same number by [latex]2[\/latex] and add\u00a0[latex]5[\/latex]. Since there is no value of [latex]x[\/latex] that will ever make this a true statement, the solution to the equation above is <i>\u201cno solution.\u201d<\/i>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]4148[\/ohm_question]\r\n\r\n<\/div>\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 \u00a0[latex]0[\/latex] satisfies the equation, so there <i>is<\/i> a solution. \u201cNo solution\u201d means that there is no value, not even\u00a0[latex]0[\/latex], 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 [latex]4[\/latex] and\u00a0[latex]5[\/latex] are values for\u00a0[latex]x[\/latex] that 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 [latex]x[\/latex] 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\u00a0[latex]3[\/latex] into the parentheses on the right-hand side.\r\n<p style=\"text-align: center;\">[latex]8y=3(y+4)+y[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\r\nNext, combine like terms.\r\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]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]4[\/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 [latex]y[\/latex] 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 [latex]2[\/latex] 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\u00a0[latex]x[\/latex] 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\u00a0[latex]10[\/latex] 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\\not=17\\end{array}[\/latex]<\/p>\r\nWe know that [latex]0\\text{ and }17[\/latex] are not equal, so there is no number that\u00a0[latex]x[\/latex] 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<h2>Algebraic Equations with an Infinite Number of Solutions<\/h2>\r\nYou have seen that if an equation has no solution, you end up with a false statement instead of a value for\u00a0[latex]x[\/latex]. It is possible to have an equation where any value for\u00a0[latex]x[\/latex] 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\u00a0[latex]x[\/latex].\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\u00a0[latex]x[\/latex] term by subtracting [latex]x[\/latex] 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 will also check to be a true statement!\r\n\r\nThis equation happens to have an infinite number of solutions. Any value for [latex]x[\/latex] 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 [latex]3[\/latex] is equal to [latex]3[\/latex] plus that same number.\u201d We know that this is always true\u2014it\u2019s the commutative property of addition!\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]184992[\/ohm_question]\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<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]x[\/latex].\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\u00a0[latex]3[\/latex] 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 [latex]x[\/latex], 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 this video, we show more examples of solving linear equations with either no solutions 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>Absolute value equations with no solutions<\/h2>\r\nAs we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance from\u00a0[latex]0[\/latex] on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]x[\/latex].\u00a0[latex]7+\\left|2x-5\\right|=4[\/latex]\r\n\r\n[reveal-answer q=\"173733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173733\"]Notice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\r\nResult of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for\u00a0[latex]x[\/latex].\u00a0 [latex]-\\frac{1}{2}\\left|x+3\\right|=6[\/latex]\r\n\r\n[reveal-answer q=\"173738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173738\"]Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\frac{1}{2}\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\frac{1}{2}\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAgain, we have a result where an absolute value is negative!\r\n\r\nThere is no solution to this equation, or DNE.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this last video, we show more\u00a0examples of absolute value equations that have no solutions.\r\n\r\nhttps:\/\/youtu.be\/T-z5cQ58I_g\r\n<h2><span style=\"color: #000000;\">Summary<\/span><\/h2>\r\n<span style=\"color: #000000;\">We have seen that solutions to equations can fall into three categories:<\/span>\r\n<ul>\r\n \t<li><span style=\"color: #000000;\">One solution.\u00a0 This is when you find the only value of the variable, such as [latex]x = 5[\/latex].<\/span><\/li>\r\n \t<li><span style=\"color: #000000;\">No solution, DNE (does not exist).\u00a0 This is when a false statement appears, like [latex]4 = 7[\/latex].<\/span><\/li>\r\n \t<li><span style=\"color: #000000;\">Many solutions, also called infinitely many solutions or All Real Numbers.\u00a0 This is when a true statement appears, like [latex]x + 3 = x + 3[\/latex].<\/span><\/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 Outcomes<\/h3>\n<ul>\n<li>Solve equations that have one solution, no solution, or an infinite number of solutions<\/li>\n<\/ul>\n<\/div>\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 [latex]x.[\/latex]\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 [latex]x[\/latex] term by subtracting [latex]2x[\/latex] 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, because [latex]4\\not=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>This is <i>not<\/i> a solution! You did <i>not<\/i> find a value for\u00a0[latex]x[\/latex]. Solving for [latex]x[\/latex] the way you know how, you arrive at the false statement [latex]4=5[\/latex]. Surely \u00a0[latex]4[\/latex] cannot be equal to [latex]5[\/latex]!<\/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 [latex]2[\/latex] and add\u00a0[latex]4[\/latex], you would never get the same answer as when you multiply that same number by [latex]2[\/latex] and add\u00a0[latex]5[\/latex]. Since there is no value of [latex]x[\/latex] that will ever make this a true statement, the solution to the equation above is <i>\u201cno solution.\u201d<\/i><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4148\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4148&theme=oea&iframe_resize_id=ohm4148&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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 \u00a0[latex]0[\/latex] satisfies the equation, so there <i>is<\/i> a solution. \u201cNo solution\u201d means that there is no value, not even\u00a0[latex]0[\/latex], 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 [latex]4[\/latex] and\u00a0[latex]5[\/latex] are values for\u00a0[latex]x[\/latex] that 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 [latex]x[\/latex] 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\u00a0[latex]3[\/latex] into the parentheses on the right-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]8y=3(y+4)+y[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\n<p>Next, combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]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]4[\/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 [latex]y[\/latex] 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 [latex]2[\/latex] 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\u00a0[latex]x[\/latex] 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\u00a0[latex]10[\/latex] 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\\not=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\u00a0[latex]x[\/latex] 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<h2>Algebraic Equations with an Infinite Number of Solutions<\/h2>\n<p>You have seen that if an equation has no solution, you end up with a false statement instead of a value for\u00a0[latex]x[\/latex]. It is possible to have an equation where any value for\u00a0[latex]x[\/latex] 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\u00a0[latex]x[\/latex].\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\u00a0[latex]x[\/latex] term by subtracting [latex]x[\/latex] 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 will also check to be a true statement!<\/p>\n<p>This equation happens to have an infinite number of solutions. Any value for [latex]x[\/latex] 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 [latex]3[\/latex] is equal to [latex]3[\/latex] plus that same number.\u201d We know that this is always true\u2014it\u2019s the commutative property of addition!<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm184992\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=184992&theme=oea&iframe_resize_id=ohm184992&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-1\" 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<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex].\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\u00a0[latex]3[\/latex] 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 [latex]x[\/latex], 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 this video, we show more examples of solving linear equations with either no solutions or many solutions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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-3\" 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>Absolute value equations with no solutions<\/h2>\n<p>As we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance from\u00a0[latex]0[\/latex] on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex].\u00a0[latex]7+\\left|2x-5\\right|=4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q173733\">Show Solution<\/span><\/p>\n<div id=\"q173733\" class=\"hidden-answer\" style=\"display: none\">Notice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\n<p>Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for\u00a0[latex]x[\/latex].\u00a0 [latex]-\\frac{1}{2}\\left|x+3\\right|=6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q173738\">Show Solution<\/span><\/p>\n<div id=\"q173738\" class=\"hidden-answer\" style=\"display: none\">Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\frac{1}{2}\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\frac{1}{2}\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Again, we have a result where an absolute value is negative!<\/p>\n<p>There is no solution to this equation, or DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this last video, we show more\u00a0examples of absolute value equations that have no solutions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Absolute Value Equations with No Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/T-z5cQ58I_g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><span style=\"color: #000000;\">Summary<\/span><\/h2>\n<p><span style=\"color: #000000;\">We have seen that solutions to equations can fall into three categories:<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">One solution.\u00a0 This is when you find the only value of the variable, such as [latex]x = 5[\/latex].<\/span><\/li>\n<li><span style=\"color: #000000;\">No solution, DNE (does not exist).\u00a0 This is when a false statement appears, like [latex]4 = 7[\/latex].<\/span><\/li>\n<li><span style=\"color: #000000;\">Many solutions, also called infinitely many solutions or All Real Numbers.\u00a0 This is when a true statement appears, like [latex]x + 3 = x + 3[\/latex].<\/span><\/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-16110\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Linear Equations with No Solutions or Infinite Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/iLkZ3o4wVxU\">https:\/\/youtu.be\/iLkZ3o4wVxU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Linear Equations with No Solutions of Infinite Solutions (Parentheses). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EU_NEo1QBJ0\">https:\/\/youtu.be\/EU_NEo1QBJ0<\/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>Absolute Value Equations with No Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/T-z5cQ58I_g\">https:\/\/youtu.be\/T-z5cQ58I_g<\/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>Beginning and Intermediate Algebra. <strong>Authored by<\/strong>: Tyler Wallace. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/wallace.ccfaculty.org\/book\/book.html\">http:\/\/wallace.ccfaculty.org\/book\/book.html<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Beginning and Intermediate Algebra\",\"author\":\"Tyler Wallace\",\"organization\":\"\",\"url\":\"http:\/\/wallace.ccfaculty.org\/book\/book.html\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"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\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Linear Equations with No 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