{"id":1248,"date":"2023-04-04T14:32:28","date_gmt":"2023-04-04T14:32:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1248"},"modified":"2024-10-18T20:51:11","modified_gmt":"2024-10-18T20:51:11","slug":"algebraic-equations-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/algebraic-equations-fresh-take\/","title":{"raw":"Algebraic Equations: Fresh Take","rendered":"Algebraic Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Simplify and calculate an algebraic equation<\/li>\r\n\t<li>Find the value of a variable that satisfies an equation<\/li>\r\n\t<li>Determine whether an equation can be solved with a single answer, cannot be solved at all, or has an infinite number of possible solutions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Algebraic Expressions<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <strong>Algebraic expressions<\/strong> are mathematical statements that combine numbers or <strong>constants<\/strong>, <strong>variables <\/strong>(letters that represent unknown numbers), and operations such as addition, subtraction, multiplication, and division. They serve as the foundational language for algebra and allow us to represent and solve complex mathematical problems. An algebraic expression can be as simple as a single variable \"[latex]x[\/latex]\", or as complex as a multi-term expression like \"[latex]3x^2 - 2x + 5[\/latex]\". <br \/>\r\n<br \/>\r\nThe parts of the expression separated by addition or subtraction are called terms, each of which can be a combination of numbers (coefficients) and variables raised to a power. To evaluate an algebraic expression, you substitute specific numerical values for the variables and perform the indicated operations. For instance, if we substitute [latex]x=2[\/latex] into the expression \"[latex]3x^2 - 2x + 5[\/latex]\", we would get [latex]3*(2)^2 - 2*2 + 5 = 12 - 4 + 5 = 13[\/latex].<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10294974&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=OF2GtIinL_s&amp;video_target=tpm-plugin-roftp0w2-OF2GtIinL_s\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Algebraic+Expressions+(Basics).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Basics)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em>\r\n<ol>\r\n\t<li>[latex]x=0[\/latex]<\/li>\r\n\t<li>[latex]x=1[\/latex]<\/li>\r\n\t<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\r\n\t<li>[latex]x=-4[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"421675\"]Show Solution[\/reveal-answer] [hidden-answer a=\"421675\"]\r\n\r\n<ol>\r\n\t<li>Substitute [latex]0[\/latex] for [latex]x[\/latex].\r\n\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(0\\right)-7 \\\\ &amp; =0-7 \\\\ &amp; =-7\\end{align}[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>Substitute [latex]1[\/latex] for [latex]x[\/latex].\r\n\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(1\\right)-7 \\\\ &amp; =2-7 \\\\ &amp; =-5\\end{align}[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\r\n\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(\\frac{1}{2}\\right)-7 \\\\ &amp; =1-7 \\\\ &amp; =-6\\end{align}[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\r\n\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(-4\\right)-7 \\\\ &amp; =-8-7 \\\\ &amp; =-15\\end{align}[\/latex]<\/div>\r\n<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>In the following video we present more examples of how to evaluate an expression for a given value.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Evaluate+Various+Algebraic+Expressions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate Various Algebraic Expressions\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Simplify Algebraic Expressions<\/h3>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> Simplifying algebraic expressions is the process of making an expression as concise as possible without changing its value. It involves several steps, often relying on mathematical properties and operations to combine like terms and eliminate unnecessary parts. <br \/>\r\n<br \/>\r\nOne fundamental process is the combination of like terms, which are terms in the expression that have the same variables and exponents. For example, in the expression [latex]3x + 2y - 5x + y[\/latex], you can combine the [latex]3x[\/latex] and [latex]-5x[\/latex] to get [latex]-2x[\/latex], and the [latex]2y[\/latex] and [latex]y[\/latex] to get [latex]3y[\/latex]. <br \/>\r\n<br \/>\r\nThus, the expression simplifies to [latex]-2x + 3y[\/latex]. The distributive property is also frequently used in simplification. It allows us to remove parentheses in expressions like [latex]3(x + 2)[\/latex], which becomes [latex]3x + 6[\/latex] after distribution.<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10294975&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=uqKY7dK_DFQ&amp;video_target=tpm-plugin-uhftq0ok-uqKY7dK_DFQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Algebraic+Expressions+(Advanced).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Advanced)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.[reveal-answer q=\"921194\"]Show Solution[\/reveal-answer] [hidden-answer a=\"921194\"]\r\n\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;P=L+W+L+W \\\\ &amp;P=L+L+W+W &amp;&amp; \\text{Commutative property of addition} \\\\ &amp;P=2L+2W &amp;&amp; \\text{Simplify} \\\\ &amp;P=2\\left(L+W\\right) &amp;&amp; \\text{Distributive property}\\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/section>\r\n<h2>Multi-Step Equations<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> Multi-step expressions refer to algebraic expressions that require multiple operations to simplify or solve. The process involves a combination of mathematical techniques including, but not limited to, applying the order of operations, combining like terms, and using properties of real numbers such as distributive, associative, and commutative properties.<\/div>\r\n<section class=\"textbox example\">Solve [latex]3y+2=11[\/latex][reveal-answer q=\"843520\"]Show Solution[\/reveal-answer] [hidden-answer a=\"843520\"]Subtract [latex]2[\/latex] from both sides of the equation to get the term with the variable by itself.\r\n\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\n\r\nDivide both sides of the equation by [latex]3[\/latex] to get a coefficient of [latex]1[\/latex] for the variable.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<h3>Solving Multi-Step Equations With Absolute Value<\/h3>\r\n<p>In the next video, we show more examples of solving a simple absolute value equation.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4g-o_-mAFpc\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+2_+Solving+Absolute+Value+Equations.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solving Absolute Value Equations\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-HrOMkIiSfU\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+4_+Solving+Absolute+Value+Equations+(Requires+Isolating+Abs.+Value).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 4: Solving Absolute Value Equations (Requires Isolating Abs. Value)\u201d here (opens in new window).<\/a><\/p>\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/2bEA7HoDfpk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+5_+Solving+Absolute+Value+Equations+(Requires+Isolating+Abs.+Value).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 5: Solving Absolute Value Equations (Requires Isolating Abs. Value)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Solving Multi-Step Equations With Parentheses<\/h3>\r\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>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/aQOkD8L57V0\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+One+Set+of+Parentheses.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving an Equation with One Set of Parentheses\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Watch the following video for a demonstration of how to solve a multi-step equation with two sets of parentheses.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/StomYTb7Xb8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+Parentheses+on+Both+Sides.txt\">transcript for \u201cSolving an Equation with Parentheses on Both Sides\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Clearing Fractions and Decimals from Equations<\/h2>\r\n<p>Watch the following video for a demonstration of how to solve a multi-step equation containing fractions by using the least common denominator to clear the fractions first.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AvJTPeACTY0\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+Fractions+(Clear+Fractions).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving an Equation with Fractions (Clear Fractions)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Watch the following example to see how to clear decimals first to solve a multi-step linear equation containing decimals.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/wtwepTZZnlY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+Decimals+(Clear+Decimals).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving an Equation with Decimals (Clear Decimals)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Equations with No Solutions<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> An equation with no solutions, also known as an inconsistent equation, is an equation that has no value for its variable that will make it a true statement. These equations often result from manipulating an original equation in a way that produces a mathematical impossibility. When you attempt to solve these types of equations, you end up with an expression that contradicts itself. For instance, an equation like [latex]2x + 3 = 2x + 5[\/latex] simplifies to [latex]3 = 5[\/latex], a statement that is always false regardless of what value [latex]x[\/latex] may have. This is an example of an equation with no solutions.<\/div>\r\n<section class=\"textbox example\">\r\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>\r\n<p><strong>a)<\/strong> Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"933839\"]Show Solution[\/reveal-answer] [hidden-answer a=\"933839\"]<br \/>\r\nFirst, distribute the [latex]3[\/latex] into the parentheses on the right-hand side.\r\n\r\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\r\n\r\nNext, begin combining like terms.\r\n\r\n<p style=\"text-align: center;\">[latex]8y=4y+12[\/latex]<\/p>\r\n\r\nNow move the variable terms to one side. Moving the [latex]4y[\/latex] will help avoid a negative sign.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,8y=4y+12\\\\\\underline{-4y\\,\\,-4y}\\\\\\,\\,\\,\\,4y=12\\end{array}[\/latex]<\/p>\r\n\r\nNow, divide each side by [latex]4y[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\Large\\frac{4y}{4}\\normalsize =\\Large\\frac{12}{4}\\normalsize\\\\y=3\\end{array}[\/latex]<\/p>\r\n\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. [\/hidden-answer]\r\n\r\n<p>&nbsp;<\/p>\r\n<p><strong>b)<\/strong> Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"937839\"]Show Solution[\/reveal-answer] [hidden-answer a=\"937839\"] First, distribute the [latex]2[\/latex] into the parentheses on the left-hand side.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}6x-10-4x=2x+7\\end{array}[\/latex]<\/p>\r\n\r\nNow begin simplifying. You can combine the [latex]x[\/latex] terms on the left-hand side.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-10=2x+7\\end{array}[\/latex]<\/p>\r\n\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 [latex]10[\/latex] be equal to that same number times two plus seven? Pretend [latex]x=3[\/latex]. 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 do not even really need to continue solving the equation, but we can just to be thorough. Add [latex]10[\/latex] to both sides.\r\n\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\n\r\nNow subtract [latex]2x[\/latex] from both sides.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,2x=2x+17\\\\\\,\\,\\underline{-2x\\,\\,-2x}\\\\\\,\\,\\,\\,\\,\\,\\,0=17\\end{array}[\/latex]<\/p>\r\n\r\nWe know that [latex]0\\text{ and }17[\/latex] are not equal, so there is no number that [latex]x[\/latex] could be to make this equation true. This false statement implies there are <strong>no solutions<\/strong> to this equation, or [latex]DNE[\/latex] (does not exist) for short. [\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Solve for [latex]x[\/latex].<center>[latex]-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6[\/latex]<\/center>[reveal-answer q=\"173738\"]Show Solution[\/reveal-answer] [hidden-answer a=\"173738\"]Notice absolute value is not alone. Multiply both sides by the reciprocal of [latex]-\\Large\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\nAgain, we have a result where an absolute value is negative! There is no solution to this equation, or [latex]DNE[\/latex]. [\/hidden-answer]<\/section>\r\n<h2>Equations with Many Solutions<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> An equation with many solutions, also known as an identity, is an equation that is true for all values of its variable. These equations often arise when both sides of an equation reduce to the same expression or if the equation is a universally true statement. For example, consider the equation [latex]3x + 2 = 3x + 2[\/latex]. No matter what value [latex]x[\/latex] is, the left and right side of the equation will always be equal. Hence, this equation is an identity and has an infinite number of solutions since all real numbers can satisfy the equation. When you attempt to solve these types of equations, you end up with an expression that is always true, such as [latex]5 = 5[\/latex]. An equation of this sort indicates that the original equation is true for any value of the variable.<\/div>\r\n<section class=\"textbox example\">Solve for [latex]x[\/latex].<center>[latex]3\\left(2x-5\\right)=6x-15[\/latex]<\/center>[reveal-answer q=\"973733\"]Show Solution[\/reveal-answer] [hidden-answer a=\"973733\"]Distribute the [latex]3[\/latex] through the parentheses on the left-hand side.\r\n\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\n\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\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\n\r\nThis true statement implies there are an infinite number of solutions to this equation. [\/hidden-answer]<\/section>\r\n<p>Watch the following video for demonstrations of equations with no solutions and infinitely many solutions.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/iLkZ3o4wVxU\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Linear+Equations+with+No+Solutions+or+Infinite+Solutions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Equations with No Solutions or Infinite Solutions\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>The next video demonstrates equations with no or infinitely many solutions involving parentheses.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/EU_NEo1QBJ0\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Linear+Equations+with+No+Solutions+of+Infinite+Solutions+(Parentheses).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Equations with No Solutions of Infinite Solutions (Parentheses)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify and calculate an algebraic equation<\/li>\n<li>Find the value of a variable that satisfies an equation<\/li>\n<li>Determine whether an equation can be solved with a single answer, cannot be solved at all, or has an infinite number of possible solutions<\/li>\n<\/ul>\n<\/section>\n<h2>Algebraic Expressions<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> <strong>Algebraic expressions<\/strong> are mathematical statements that combine numbers or <strong>constants<\/strong>, <strong>variables <\/strong>(letters that represent unknown numbers), and operations such as addition, subtraction, multiplication, and division. They serve as the foundational language for algebra and allow us to represent and solve complex mathematical problems. An algebraic expression can be as simple as a single variable &#8220;[latex]x[\/latex]&#8220;, or as complex as a multi-term expression like &#8220;[latex]3x^2 - 2x + 5[\/latex]&#8220;. <\/p>\n<p>The parts of the expression separated by addition or subtraction are called terms, each of which can be a combination of numbers (coefficients) and variables raised to a power. To evaluate an algebraic expression, you substitute specific numerical values for the variables and perform the indicated operations. For instance, if we substitute [latex]x=2[\/latex] into the expression &#8220;[latex]3x^2 - 2x + 5[\/latex]&#8220;, we would get [latex]3*(2)^2 - 2*2 + 5 = 12 - 4 + 5 = 13[\/latex].<\/p><\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10294974&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=OF2GtIinL_s&amp;video_target=tpm-plugin-roftp0w2-OF2GtIinL_s\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Algebraic+Expressions+(Basics).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Basics)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em><\/p>\n<ol>\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]x=-4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q421675\">Show Solution<\/button> <\/p>\n<div id=\"q421675\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]0[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(0\\right)-7 \\\\ & =0-7 \\\\ & =-7\\end{align}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<li>Substitute [latex]1[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(1\\right)-7 \\\\ & =2-7 \\\\ & =-5\\end{align}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(\\frac{1}{2}\\right)-7 \\\\ & =1-7 \\\\ & =-6\\end{align}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(-4\\right)-7 \\\\ & =-8-7 \\\\ & =-15\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video we present more examples of how to evaluate an expression for a given value.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Evaluate+Various+Algebraic+Expressions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate Various Algebraic Expressions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Simplify Algebraic Expressions<\/h3>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> Simplifying algebraic expressions is the process of making an expression as concise as possible without changing its value. It involves several steps, often relying on mathematical properties and operations to combine like terms and eliminate unnecessary parts. <\/p>\n<p>One fundamental process is the combination of like terms, which are terms in the expression that have the same variables and exponents. For example, in the expression [latex]3x + 2y - 5x + y[\/latex], you can combine the [latex]3x[\/latex] and [latex]-5x[\/latex] to get [latex]-2x[\/latex], and the [latex]2y[\/latex] and [latex]y[\/latex] to get [latex]3y[\/latex]. <\/p>\n<p>Thus, the expression simplifies to [latex]-2x + 3y[\/latex]. The distributive property is also frequently used in simplification. It allows us to remove parentheses in expressions like [latex]3(x + 2)[\/latex], which becomes [latex]3x + 6[\/latex] after distribution.<\/p><\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10294975&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=uqKY7dK_DFQ&amp;video_target=tpm-plugin-uhftq0ok-uqKY7dK_DFQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Algebraic+Expressions+(Advanced).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Advanced)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q921194\">Show Solution<\/button> <\/p>\n<div id=\"q921194\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{align}&P=L+W+L+W \\\\ &P=L+L+W+W && \\text{Commutative property of addition} \\\\ &P=2L+2W && \\text{Simplify} \\\\ &P=2\\left(L+W\\right) && \\text{Distributive property}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Multi-Step Equations<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> Multi-step expressions refer to algebraic expressions that require multiple operations to simplify or solve. The process involves a combination of mathematical techniques including, but not limited to, applying the order of operations, combining like terms, and using properties of real numbers such as distributive, associative, and commutative properties.<\/div>\n<section class=\"textbox example\">Solve [latex]3y+2=11[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q843520\">Show Solution<\/button> <\/p>\n<div id=\"q843520\" class=\"hidden-answer\" style=\"display: none\">Subtract [latex]2[\/latex] 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 [latex]3[\/latex] to get a coefficient of [latex]1[\/latex] for the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3>Solving Multi-Step Equations With Absolute Value<\/h3>\n<p>In the next video, we show more examples of solving a simple absolute value equation.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4g-o_-mAFpc\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+2_+Solving+Absolute+Value+Equations.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Solving Absolute Value Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-HrOMkIiSfU\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+4_+Solving+Absolute+Value+Equations+(Requires+Isolating+Abs.+Value).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 4: Solving Absolute Value Equations (Requires Isolating Abs. Value)\u201d here (opens in new window).<\/a><\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/2bEA7HoDfpk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+5_+Solving+Absolute+Value+Equations+(Requires+Isolating+Abs.+Value).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 5: Solving Absolute Value Equations (Requires Isolating Abs. Value)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Solving Multi-Step Equations With Parentheses<\/h3>\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<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/aQOkD8L57V0\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+One+Set+of+Parentheses.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving an Equation with One Set of Parentheses\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video for a demonstration of how to solve a multi-step equation with two sets of parentheses.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/StomYTb7Xb8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+Parentheses+on+Both+Sides.txt\">transcript for \u201cSolving an Equation with Parentheses on Both Sides\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Clearing Fractions and Decimals from Equations<\/h2>\n<p>Watch the following video for a demonstration of how to solve a multi-step equation containing fractions by using the least common denominator to clear the fractions first.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/AvJTPeACTY0\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+Fractions+(Clear+Fractions).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving an Equation with Fractions (Clear Fractions)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following example to see how to clear decimals first to solve a multi-step linear equation containing decimals.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/wtwepTZZnlY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+an+Equation+with+Decimals+(Clear+Decimals).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving an Equation with Decimals (Clear Decimals)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Equations with No Solutions<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> An equation with no solutions, also known as an inconsistent equation, is an equation that has no value for its variable that will make it a true statement. These equations often result from manipulating an original equation in a way that produces a mathematical impossibility. When you attempt to solve these types of equations, you end up with an expression that contradicts itself. For instance, an equation like [latex]2x + 3 = 2x + 5[\/latex] simplifies to [latex]3 = 5[\/latex], a statement that is always false regardless of what value [latex]x[\/latex] may have. This is an example of an equation with no solutions.<\/div>\n<section class=\"textbox example\">\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><strong>a)<\/strong> Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q933839\">Show Solution<\/button> <\/p>\n<div id=\"q933839\" class=\"hidden-answer\" style=\"display: none\">\nFirst, distribute the [latex]3[\/latex] into the parentheses on the right-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\n<p>Next, begin combining like terms.<\/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]4y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\Large\\frac{4y}{4}\\normalsize =\\Large\\frac{12}{4}\\normalsize\\\\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><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong>b)<\/strong> Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q937839\">Show Solution<\/button> <\/p>\n<div id=\"q937839\" class=\"hidden-answer\" style=\"display: none\"> 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}6x-10-4x=2x+7\\end{array}[\/latex]<\/p>\n<p>Now begin simplifying. You can combine the [latex]x[\/latex] terms on the left-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}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 [latex]10[\/latex] be equal to that same number times two plus seven? Pretend [latex]x=3[\/latex]. 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 do not even really need to continue solving the equation, but we can just to be thorough. 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 subtract [latex]2x[\/latex] from both sides.<\/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 [latex]x[\/latex] could be to make this equation true. This false statement implies there are <strong>no solutions<\/strong> to this equation, or [latex]DNE[\/latex] (does not exist) for short. <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Solve for [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q173738\">Show Solution<\/button> <\/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]-\\Large\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\Large\\frac{1}{2}\\normalsize\\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! There is no solution to this equation, or [latex]DNE[\/latex]. <\/p><\/div>\n<\/div>\n<\/section>\n<h2>Equations with Many Solutions<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> An equation with many solutions, also known as an identity, is an equation that is true for all values of its variable. These equations often arise when both sides of an equation reduce to the same expression or if the equation is a universally true statement. For example, consider the equation [latex]3x + 2 = 3x + 2[\/latex]. No matter what value [latex]x[\/latex] is, the left and right side of the equation will always be equal. Hence, this equation is an identity and has an infinite number of solutions since all real numbers can satisfy the equation. When you attempt to solve these types of equations, you end up with an expression that is always true, such as [latex]5 = 5[\/latex]. An equation of this sort indicates that the original equation is true for any value of the variable.<\/div>\n<section class=\"textbox example\">Solve for [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]3\\left(2x-5\\right)=6x-15[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q973733\">Show Solution<\/button> <\/p>\n<div id=\"q973733\" class=\"hidden-answer\" style=\"display: none\">Distribute the [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><\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for demonstrations of equations with no solutions and infinitely many solutions.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/iLkZ3o4wVxU\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Linear+Equations+with+No+Solutions+or+Infinite+Solutions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Equations with No Solutions or Infinite Solutions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The next video demonstrates equations with no or infinitely many solutions involving parentheses.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/EU_NEo1QBJ0\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Linear+Equations+with+No+Solutions+of+Infinite+Solutions+(Parentheses).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Equations with No Solutions of Infinite Solutions (Parentheses)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":33,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1248"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":35,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1248\/revisions"}],"predecessor-version":[{"id":15273,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1248\/revisions\/15273"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1248\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1248"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1248"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1248"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}