{"id":8335,"date":"2017-05-01T22:48:06","date_gmt":"2017-05-01T22:48:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=8335"},"modified":"2018-01-10T19:11:51","modified_gmt":"2018-01-10T19:11:51","slug":"solve-a-formula-for-a-specific-variable","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/solve-a-formula-for-a-specific-variable\/","title":{"raw":"Solving a Formula for a Specific Variable","rendered":"Solving a Formula for a Specific Variable"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve a formula or equation for a specific variable using the properties of equality<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences\u2014fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft Excel<sup>TM<\/sup> relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.\r\n\r\nIn some examples, we used the formula [latex]d=rt[\/latex]. This formula gives the value of [latex]d[\/latex] when you substitute in the values of [latex]r[\/latex] and [latex]t[\/latex]. But in another example, we had to find the value of [latex]t[\/latex]. We substituted in values of [latex]d[\/latex] and [latex]r[\/latex] and then used algebra to solve for [latex]t[\/latex]. If you had to do this often, you might wonder why there isn\u2019t a formula that gives the value of [latex]t[\/latex] when you substitute in the values of [latex]d[\/latex] and [latex]r[\/latex]. We can get a formula like this by solving the formula [latex]d=rt[\/latex] for [latex]t[\/latex].\r\n<div class=\"textbox shaded\"><strong>To solve a formula for a specific variable<\/strong> means to get that variable by itself with a coefficient of [latex]1[\/latex] on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable <em>in general.<\/em> This process is also called <em>solving a literal equation<\/em>. The result is another formula, made up only of variables. The formula contains letters, or <em>literals<\/em>.<\/div>\r\nLet\u2019s try a few examples, starting with the distance, rate, and time formula we used above.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve the formula [latex]d=rt[\/latex] for [latex]t\\text{:}[\/latex]\r\n<ol>\r\n \t<li>When [latex]d=520[\/latex] and [latex]r=65[\/latex]<\/li>\r\n \t<li>In general.<\/li>\r\n<\/ol>\r\nSolution:\r\nWe\u2019ll write the solutions side-by-side so you can see that solving a formula in general uses the same steps as when we have numbers to substitute.\r\n<table id=\"eip-id1164150753614\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1. When [latex]d = 520[\/latex] and [latex]r = 65[\/latex]<\/td>\r\n<td>2. In general<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the formula.<\/td>\r\n<td>[latex]d=rt[\/latex]<\/td>\r\n<td>[latex]d=rt[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute any given values.<\/td>\r\n<td>[latex]520=65t[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide to isolate <em>t<\/em>.<\/td>\r\n<td>[latex]{\\Large\\frac{520}{65}}={\\Large\\frac{65t}{65}}[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{d}{r}}={\\Large\\frac{rt}{r}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]8=t[\/latex]\r\n\r\n[latex]t=8[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{d}{r}}=t[\/latex]\r\n\r\n[latex]t={\\Large\\frac{d}{r}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nWe say the formula [latex]t={\\Large\\frac{d}{r}}[\/latex] is solved for [latex]t[\/latex]. We can use this version of the formula any time we are given the distance and rate and need to find the time.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n<iframe id=\"mom225\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145634&amp;theme=oea&amp;iframe_resize_id=mom225\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nWe used the formula [latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex] to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe formula for area of a triangle is [latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex]. Solve this formula for [latex]h\\text{:}[\/latex]\r\n<ol>\r\n \t<li>When [latex]A=90[\/latex] and [latex]b=15[\/latex]<\/li>\r\n \t<li>In general<\/li>\r\n<\/ol>\r\n<p class=\"p1\">[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"190834\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1170572798895\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1. When <em>A<\/em> = 90 and <em>b<\/em> = 15<\/td>\r\n<td>2. In general<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the forumla.<\/td>\r\n<td>[latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex]<\/td>\r\n<td>[latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute any given values.<\/td>\r\n<td>[latex]90=\\Large\\frac{1}{2}\\normalsize\\cdot{15}\\cdot{h}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Clear the fractions.<\/td>\r\n<td>[latex]\\color{red}{2}\\cdot{90}=\\color{red}{2}\\cdot\\Large\\frac{1}{2}\\normalsize\\cdot{15}\\cdot{h}[\/latex]<\/td>\r\n<td>[latex]\\color{red}{2}\\cdot{A}=\\color{red}{2}\\cdot\\Large\\frac{1}{2}\\normalsize\\cdot{b}\\cdot{h}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]180=15h[\/latex]<\/td>\r\n<td>[latex]2A=bh[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Solve for <em>h<\/em>.<\/td>\r\n<td>[latex]12=h[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{2A}{b}}=h[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can now find the height of a triangle, if we know the area and the base, by using the formula\r\n\r\n[latex]h={\\Large\\frac{2A}{b}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom700\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145635&amp;theme=oea&amp;iframe_resize_id=mom700\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nPreviously, we used the formula [latex]I=Prt[\/latex] to calculate simple interest, where [latex]I[\/latex] is interest, [latex]P[\/latex] is principal, [latex]r[\/latex] is rate as a decimal, and [latex]t[\/latex] is time in years.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve the formula [latex]I=Prt[\/latex] to find the principal, [latex]P\\text{:}[\/latex]\r\n<ol>\r\n \t<li>When [latex]I=\\text{\\$5,600},r=\\text{4%},t=7\\text{years}[\/latex]<\/li>\r\n \t<li>In general<\/li>\r\n<\/ol>\r\n<p class=\"p1\">[reveal-answer q=\"542986\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"542986\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168058902092\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><em>1. \u00a0I<\/em> = $5600, <em>r<\/em> = 4%, <em>t<\/em> = 7 years<\/td>\r\n<td>2. In general<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the forumla.<\/td>\r\n<td>[latex]I=Prt[\/latex]<\/td>\r\n<td>[latex]I=Prt[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute any given values.<\/td>\r\n<td>[latex]5600=P(0.04)(7)[\/latex]<\/td>\r\n<td>[latex]I=Prt[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply <em>r<\/em> \u22c5 <em>t<\/em>.<\/td>\r\n<td>[latex]5600=P(0.28)[\/latex]<\/td>\r\n<td>[latex]I=P(rt)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide to isolate <em>P<\/em>.<\/td>\r\n<td>[latex]\\Large\\frac{5600}{\\color{red}{0.28}}\\normalsize =\\Large\\frac{P(0.28)}{\\color{red}{0.28}}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{I}{\\color{red}{rt}}\\normalsize =\\Large\\frac{P(rt)}{\\color{red}{rt}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]20,000=P[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{I}{rt}\\normalsize =P[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>State the answer.<\/td>\r\n<td>The principal is $20,000.<\/td>\r\n<td>[latex]P=\\Large\\frac{I}{rt}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom900\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145640&amp;theme=oea&amp;iframe_resize_id=mom900\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nWatch the following video to see another\u00a0example of how to solve an equation for a specific variable.\r\n\r\nhttps:\/\/youtu.be\/VQZQvJ3rXYg\r\n\r\nLater in this class, and in future algebra classes, you\u2019ll encounter equations that relate two variables, usually [latex]x[\/latex] and [latex]y[\/latex]. You might be given an equation that is solved for [latex]y[\/latex] and you need to solve it for [latex]x[\/latex], or vice versa. In the following example, we\u2019re given an equation with both [latex]x[\/latex] and [latex]y[\/latex] on the same side and we\u2019ll solve it for [latex]y[\/latex]. To do this, we will follow the same steps that we used to solve a formula for a specific variable.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve the formula [latex]3x+2y=18[\/latex] for [latex]y\\text{:}[\/latex]\r\n<ol>\r\n \t<li>When [latex]x=4[\/latex]<\/li>\r\n \t<li>In general<\/li>\r\n<\/ol>\r\n<p class=\"p1\">[reveal-answer q=\"908211\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"908211\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1172082377084\" class=\"unnumbered unstyled\" summary=\"A table is shown. The columns are labeled \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>1. When <em>x<\/em> = 4<\/td>\r\n<td>2. In general<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the equation.<\/td>\r\n<td>[latex]3x+2y=18[\/latex]<\/td>\r\n<td>[latex]3x+2y=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute any given values.<\/td>\r\n<td>[latex]3(4)+2y=18[\/latex]<\/td>\r\n<td>[latex]3x+2y=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify if possible.<\/td>\r\n<td>[latex]12+2y=18[\/latex]<\/td>\r\n<td>[latex]3x+2y=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to isolate the <em>y<\/em>-term.<\/td>\r\n<td>[latex]12\\color{red}{-12}+2y=18\\color{red}{-12}[\/latex]<\/td>\r\n<td>[latex]3x\\color{red}{-3x}+2y=18\\color{red}{-3x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]2y=6[\/latex]<\/td>\r\n<td>[latex]2y=18-3x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]\\Large\\frac{2y}{\\color{red}{2}}\\normalsize =\\Large\\frac{6}{\\color{red}{2}}[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{2y}{\\color{red}{2}}\\normalsize =\\Large\\frac{18-3x}{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=3[\/latex]<\/td>\r\n<td>[latex]y=\\Large\\frac{18-3x}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n&nbsp;\r\n\r\n<iframe id=\"mom220\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142912&amp;theme=oea&amp;iframe_resize_id=mom220\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you\u2019re ready to solve a formula in general without using numbers as a guide?\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve the formula [latex]P=a+b+c[\/latex] for [latex]a[\/latex].\r\n<p class=\"p1\">[reveal-answer q=\"872233\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"872233\"]<\/p>\r\nSolution:\r\nWe will isolate [latex]a[\/latex] on one side of the equation.\r\n<table id=\"eip-id1168469748609\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td>We will isolate <em>a<\/em> on one side of the equation.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the equation.<\/td>\r\n<td><\/td>\r\n<td>[latex]P=a+b+c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract <em>b<\/em> and <em>c<\/em> from both sides to isolate <em>a<\/em>.<\/td>\r\n<td><\/td>\r\n<td>[latex]P\\color{red}{-b-c}=a+b+c\\color{red}{-b-c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td><\/td>\r\n<td>[latex]P-b-c=a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo, [latex]a=P-b-c[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n&nbsp;\r\n\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142894&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve the equation [latex]3x+y=10[\/latex] for [latex]y[\/latex].\r\n<p class=\"p1\">[reveal-answer q=\"923766\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"923766\"]<\/p>\r\nSolution\r\nWe will isolate [latex]y[\/latex] on one side of the equation.\r\n<table id=\"eip-id1168468686086\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td>We will isolate <em>y<\/em> on one side of the equation.<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the equation.<\/td>\r\n<td><\/td>\r\n<td>[latex]3x+y=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract 3<em>x<\/em> from both sides to isolate <em>y<\/em>.<\/td>\r\n<td><\/td>\r\n<td>[latex]3x\\color{red}{-3x}+y=10\\color{red}{-3x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td><\/td>\r\n<td>[latex]y=10 - 3x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom30\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142892&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve the equation [latex]6x+5y=13[\/latex] for [latex]y[\/latex].\r\n<p class=\"p1\">[reveal-answer q=\"259834\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"259834\"]<\/p>\r\nSolution:\r\nWe will isolate [latex]y[\/latex] on one side of the equation.\r\n<table id=\"eip-id1168469523651\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td>We will isolate <em>y<\/em> on one side of the equation.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the equation.<\/td>\r\n<td>[latex]6x+5y=13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to isolate the term with <em>y<\/em>.<\/td>\r\n<td>[latex]6x+5y\\color{red}{-6x}=13\\color{red}{-6x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]5y=13-6x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide by 5 to make the coefficient 1.<\/td>\r\n<td>[latex]\\Large\\frac{5y}{\\color{red}{5}}\\normalsize =\\Large\\frac{13-6x}{\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=\\Large\\frac{13-6x}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom40\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142895&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we show another example of how to solve an equation for a specific variable.\r\n\r\nhttps:\/\/youtu.be\/dG0_i8lN2y0","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve a formula or equation for a specific variable using the properties of equality<\/li>\n<\/ul>\n<\/div>\n<p>In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences\u2014fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft Excel<sup>TM<\/sup> relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.<\/p>\n<p>In some examples, we used the formula [latex]d=rt[\/latex]. This formula gives the value of [latex]d[\/latex] when you substitute in the values of [latex]r[\/latex] and [latex]t[\/latex]. But in another example, we had to find the value of [latex]t[\/latex]. We substituted in values of [latex]d[\/latex] and [latex]r[\/latex] and then used algebra to solve for [latex]t[\/latex]. If you had to do this often, you might wonder why there isn\u2019t a formula that gives the value of [latex]t[\/latex] when you substitute in the values of [latex]d[\/latex] and [latex]r[\/latex]. We can get a formula like this by solving the formula [latex]d=rt[\/latex] for [latex]t[\/latex].<\/p>\n<div class=\"textbox shaded\"><strong>To solve a formula for a specific variable<\/strong> means to get that variable by itself with a coefficient of [latex]1[\/latex] on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable <em>in general.<\/em> This process is also called <em>solving a literal equation<\/em>. The result is another formula, made up only of variables. The formula contains letters, or <em>literals<\/em>.<\/div>\n<p>Let\u2019s try a few examples, starting with the distance, rate, and time formula we used above.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve the formula [latex]d=rt[\/latex] for [latex]t\\text{:}[\/latex]<\/p>\n<ol>\n<li>When [latex]d=520[\/latex] and [latex]r=65[\/latex]<\/li>\n<li>In general.<\/li>\n<\/ol>\n<p>Solution:<br \/>\nWe\u2019ll write the solutions side-by-side so you can see that solving a formula in general uses the same steps as when we have numbers to substitute.<\/p>\n<table id=\"eip-id1164150753614\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>1. When [latex]d = 520[\/latex] and [latex]r = 65[\/latex]<\/td>\n<td>2. In general<\/td>\n<\/tr>\n<tr>\n<td>Write the formula.<\/td>\n<td>[latex]d=rt[\/latex]<\/td>\n<td>[latex]d=rt[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute any given values.<\/td>\n<td>[latex]520=65t[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Divide to isolate <em>t<\/em>.<\/td>\n<td>[latex]{\\Large\\frac{520}{65}}={\\Large\\frac{65t}{65}}[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{d}{r}}={\\Large\\frac{rt}{r}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]8=t[\/latex]<\/p>\n<p>[latex]t=8[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{d}{r}}=t[\/latex]<\/p>\n<p>[latex]t={\\Large\\frac{d}{r}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>We say the formula [latex]t={\\Large\\frac{d}{r}}[\/latex] is solved for [latex]t[\/latex]. We can use this version of the formula any time we are given the distance and rate and need to find the time.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom225\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145634&amp;theme=oea&amp;iframe_resize_id=mom225\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>We used the formula [latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex] to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The formula for area of a triangle is [latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex]. Solve this formula for [latex]h\\text{:}[\/latex]<\/p>\n<ol>\n<li>When [latex]A=90[\/latex] and [latex]b=15[\/latex]<\/li>\n<li>In general<\/li>\n<\/ol>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1170572798895\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>1. When <em>A<\/em> = 90 and <em>b<\/em> = 15<\/td>\n<td>2. In general<\/td>\n<\/tr>\n<tr>\n<td>Write the forumla.<\/td>\n<td>[latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex]<\/td>\n<td>[latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute any given values.<\/td>\n<td>[latex]90=\\Large\\frac{1}{2}\\normalsize\\cdot{15}\\cdot{h}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Clear the fractions.<\/td>\n<td>[latex]\\color{red}{2}\\cdot{90}=\\color{red}{2}\\cdot\\Large\\frac{1}{2}\\normalsize\\cdot{15}\\cdot{h}[\/latex]<\/td>\n<td>[latex]\\color{red}{2}\\cdot{A}=\\color{red}{2}\\cdot\\Large\\frac{1}{2}\\normalsize\\cdot{b}\\cdot{h}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]180=15h[\/latex]<\/td>\n<td>[latex]2A=bh[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Solve for <em>h<\/em>.<\/td>\n<td>[latex]12=h[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{2A}{b}}=h[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can now find the height of a triangle, if we know the area and the base, by using the formula<\/p>\n<p>[latex]h={\\Large\\frac{2A}{b}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom700\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145635&amp;theme=oea&amp;iframe_resize_id=mom700\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>Previously, we used the formula [latex]I=Prt[\/latex] to calculate simple interest, where [latex]I[\/latex] is interest, [latex]P[\/latex] is principal, [latex]r[\/latex] is rate as a decimal, and [latex]t[\/latex] is time in years.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve the formula [latex]I=Prt[\/latex] to find the principal, [latex]P\\text{:}[\/latex]<\/p>\n<ol>\n<li>When [latex]I=\\text{\\$5,600},r=\\text{4%},t=7\\text{years}[\/latex]<\/li>\n<li>In general<\/li>\n<\/ol>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q542986\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q542986\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168058902092\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><em>1. \u00a0I<\/em> = $5600, <em>r<\/em> = 4%, <em>t<\/em> = 7 years<\/td>\n<td>2. In general<\/td>\n<\/tr>\n<tr>\n<td>Write the forumla.<\/td>\n<td>[latex]I=Prt[\/latex]<\/td>\n<td>[latex]I=Prt[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute any given values.<\/td>\n<td>[latex]5600=P(0.04)(7)[\/latex]<\/td>\n<td>[latex]I=Prt[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply <em>r<\/em> \u22c5 <em>t<\/em>.<\/td>\n<td>[latex]5600=P(0.28)[\/latex]<\/td>\n<td>[latex]I=P(rt)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide to isolate <em>P<\/em>.<\/td>\n<td>[latex]\\Large\\frac{5600}{\\color{red}{0.28}}\\normalsize =\\Large\\frac{P(0.28)}{\\color{red}{0.28}}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{I}{\\color{red}{rt}}\\normalsize =\\Large\\frac{P(rt)}{\\color{red}{rt}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]20,000=P[\/latex]<\/td>\n<td>[latex]\\Large\\frac{I}{rt}\\normalsize =P[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>State the answer.<\/td>\n<td>The principal is $20,000.<\/td>\n<td>[latex]P=\\Large\\frac{I}{rt}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom900\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145640&amp;theme=oea&amp;iframe_resize_id=mom900\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see another\u00a0example of how to solve an equation for a specific variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Find the Base of a Triangle Given Area \/ Literal Equation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VQZQvJ3rXYg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Later in this class, and in future algebra classes, you\u2019ll encounter equations that relate two variables, usually [latex]x[\/latex] and [latex]y[\/latex]. You might be given an equation that is solved for [latex]y[\/latex] and you need to solve it for [latex]x[\/latex], or vice versa. In the following example, we\u2019re given an equation with both [latex]x[\/latex] and [latex]y[\/latex] on the same side and we\u2019ll solve it for [latex]y[\/latex]. To do this, we will follow the same steps that we used to solve a formula for a specific variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve the formula [latex]3x+2y=18[\/latex] for [latex]y\\text{:}[\/latex]<\/p>\n<ol>\n<li>When [latex]x=4[\/latex]<\/li>\n<li>In general<\/li>\n<\/ol>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q908211\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q908211\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1172082377084\" class=\"unnumbered unstyled\" summary=\"A table is shown. The columns are labeled\">\n<tbody>\n<tr>\n<td><\/td>\n<td>1. When <em>x<\/em> = 4<\/td>\n<td>2. In general<\/td>\n<\/tr>\n<tr>\n<td>Write the equation.<\/td>\n<td>[latex]3x+2y=18[\/latex]<\/td>\n<td>[latex]3x+2y=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute any given values.<\/td>\n<td>[latex]3(4)+2y=18[\/latex]<\/td>\n<td>[latex]3x+2y=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify if possible.<\/td>\n<td>[latex]12+2y=18[\/latex]<\/td>\n<td>[latex]3x+2y=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to isolate the <em>y<\/em>-term.<\/td>\n<td>[latex]12\\color{red}{-12}+2y=18\\color{red}{-12}[\/latex]<\/td>\n<td>[latex]3x\\color{red}{-3x}+2y=18\\color{red}{-3x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]2y=6[\/latex]<\/td>\n<td>[latex]2y=18-3x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]\\Large\\frac{2y}{\\color{red}{2}}\\normalsize =\\Large\\frac{6}{\\color{red}{2}}[\/latex]<\/td>\n<td>[latex]\\Large\\frac{2y}{\\color{red}{2}}\\normalsize =\\Large\\frac{18-3x}{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=3[\/latex]<\/td>\n<td>[latex]y=\\Large\\frac{18-3x}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"mom220\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142912&amp;theme=oea&amp;iframe_resize_id=mom220\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you\u2019re ready to solve a formula in general without using numbers as a guide?<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve the formula [latex]P=a+b+c[\/latex] for [latex]a[\/latex].<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q872233\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q872233\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe will isolate [latex]a[\/latex] on one side of the equation.<\/p>\n<table id=\"eip-id1168469748609\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>We will isolate <em>a<\/em> on one side of the equation.<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write the equation.<\/td>\n<td><\/td>\n<td>[latex]P=a+b+c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract <em>b<\/em> and <em>c<\/em> from both sides to isolate <em>a<\/em>.<\/td>\n<td><\/td>\n<td>[latex]P\\color{red}{-b-c}=a+b+c\\color{red}{-b-c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><\/td>\n<td>[latex]P-b-c=a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So, [latex]a=P-b-c[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142894&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve the equation [latex]3x+y=10[\/latex] for [latex]y[\/latex].<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q923766\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q923766\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe will isolate [latex]y[\/latex] on one side of the equation.<\/p>\n<table id=\"eip-id1168468686086\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>We will isolate <em>y<\/em> on one side of the equation.<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write the equation.<\/td>\n<td><\/td>\n<td>[latex]3x+y=10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract 3<em>x<\/em> from both sides to isolate <em>y<\/em>.<\/td>\n<td><\/td>\n<td>[latex]3x\\color{red}{-3x}+y=10\\color{red}{-3x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><\/td>\n<td>[latex]y=10 - 3x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom30\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142892&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve the equation [latex]6x+5y=13[\/latex] for [latex]y[\/latex].<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q259834\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q259834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe will isolate [latex]y[\/latex] on one side of the equation.<\/p>\n<table id=\"eip-id1168469523651\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>We will isolate <em>y<\/em> on one side of the equation.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write the equation.<\/td>\n<td>[latex]6x+5y=13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to isolate the term with <em>y<\/em>.<\/td>\n<td>[latex]6x+5y\\color{red}{-6x}=13\\color{red}{-6x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]5y=13-6x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide by 5 to make the coefficient 1.<\/td>\n<td>[latex]\\Large\\frac{5y}{\\color{red}{5}}\\normalsize =\\Large\\frac{13-6x}{\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=\\Large\\frac{13-6x}{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom40\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142895&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to solve an equation for a specific variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Literal Equations:  Solve ax-by=c for y\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/dG0_i8lN2y0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-8335\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Find the Base of a Triangle Given Area \/ Literal Equation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/VQZQvJ3rXYg\">https:\/\/youtu.be\/VQZQvJ3rXYg<\/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>Literal Equations: Solve ax-by=c for y. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/dG0_i8lN2y0\">https:\/\/youtu.be\/dG0_i8lN2y0<\/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 142912, 142894, 142892, 145640, 142895, 145635. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 142895. <strong>Authored by<\/strong>: Sousa James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Find the Base of a Triangle Given Area \/ Literal Equation\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/VQZQvJ3rXYg\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Literal Equations: Solve ax-by=c for 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