{"id":99,"date":"2023-06-21T13:22:33","date_gmt":"2023-06-21T13:22:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/writing-a-linear-equation-to-solve-an-application\/"},"modified":"2023-09-07T15:45:48","modified_gmt":"2023-09-07T15:45:48","slug":"writing-a-linear-equation-to-solve-an-application","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/writing-a-linear-equation-to-solve-an-application\/","title":{"raw":"\u25aa   Writing a Linear Equation to Solve an Application*","rendered":"\u25aa   Writing a Linear Equation to Solve an Application*"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write a linear equation to express the relationship between unknown quantities.<\/li>\r\n \t<li>Write a linear equation to model a real-world situation.<\/li>\r\n \t<li>Write a linear equation in one variable to solve problems with two unknowns.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall expressions and equations<\/h3>\r\nRecall that a mathematical\u00a0<strong>expression<\/strong>\u00a0consists of <strong>terms<\/strong> connected by addition or subtraction, each term of which consists of <strong>variables<\/strong> and numbers connected by multiplication or division. A number that multiplies a variable, such as the [latex]2[\/latex] in the term [latex]2x[\/latex] is called the\u00a0<strong>coefficient\u00a0<\/strong>of the variable.\r\n\r\nAn\u00a0<strong>equation<\/strong> is a mathematical statement of the equivalency of two expressions, one on either side of an equal sign.\r\n\r\n<strong>Expressions<\/strong> may be combined or simplified by rearranging or doing operations on the terms. <strong>Equations<\/strong> may be solved for the value or values of the variable that make the statement of equivalency true.\r\n\r\n<\/div>\r\nConsider a car rental agency that charges $150 per week plus $0.10 per mile driven to rent a compact car. We can use these quantities to write an equation that models the cost of renting the car for a week [latex]C[\/latex] given a certain number of miles [latex]x[\/latex] driven.\r\n<div style=\"text-align: center;\">[latex]C=150+0.10x[\/latex]<\/div>\r\nTo set up a linear equation that models a real-world situation, we must first determine the known quantities and define the unknown quantity as a variable. Then, we interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10 per mile, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write [latex]0.10x[\/latex] to model the portion of the weekly cost generated by miles driven. This expression represents a variable cost because it changes according to the number of miles driven.\r\n\r\nIf a quantity is independent of a variable, we usually just add or subtract it according to the problem. In the car rental example, there is a flat fee of $150 to rent the car, independent of the number of miles driven. In applications involving costs, amounts such as this flat fee that do not change are often called fixed costs.\r\n\r\nWhen dealing with real-world applications, there are certain expressions that we can translate directly into math. The table below lists some common verbal expressions and their equivalent mathematical expressions.\r\n<table summary=\"A table with 8 rows and 2 columns. The entries in the first row are: Verbal and Translation to math operations. The entries in the second row are: One number exceeds another by a and x, x+a. The entries in the third row are: Twice a number and 2x. The entries in the fourth row are: One number is a more than another number and x, x plus a. The entries in the fifth row are: One number is a less than twice another number and x,2 times x minus a. The entries in the sixth row are: The product of a number and a, decreased by b and a times x minus b. The entries in the seventh row are: The quotient of a number and the number plus a is three times the number and x divided by the quantity x plus a equals three times x. The entries in the eighth row are: The product of three times a number and the number decreased by b is c and three times x times the quantity x minus b equals c.\">\r\n<thead>\r\n<tr>\r\n<th>Verbal<\/th>\r\n<th>Translation to Math Operations<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>One number exceeds another by <em>a<\/em><\/td>\r\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Twice a number<\/td>\r\n<td>[latex]2x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>One number is <em>a <\/em>more than another number<\/td>\r\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>One number is <em>a <\/em>less than twice another number<\/td>\r\n<td>[latex]x,2x-a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product of a number and <em>a<\/em>, decreased by <em>b<\/em><\/td>\r\n<td>[latex]ax-b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The quotient of a number and the number plus <em>a <\/em>is three times the number<\/td>\r\n<td>[latex]\\Large\\frac{x}{x+a}\\normalsize =3x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product of three times a number and the number decreased by <em>b <\/em>is <em>c<\/em><\/td>\r\n<td>[latex]3x\\left(x-b\\right)=c[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the car-rental example above, we identified the unknown quantity (number of miles driven), and assigned it the variable [latex]x[\/latex]. Then we identified known quantities, and translated the words in the situation into relationships between the known and unknown quantities to write an equation that models the situation. We could use the model to answer questions about the situation such as finding the total cost for a week if 500 miles were driven or how many miles could be driven in a week on a budget of $375.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a real-world situation, write a linear equation to model it<\/h3>\r\n<ol>\r\n \t<li>Identify known and unknown quantities.<\/li>\r\n \t<li>Assign a variable to represent the unknown quantity.<\/li>\r\n \t<li>If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.<\/li>\r\n \t<li>Write an equation interpreting the words in the problem as mathematical operations.<\/li>\r\n \t<li>Solve the equation, check to be sure your answer is reasonable, and give the answer using the language and units of the original situation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind a linear equation to solve for the following unknown quantities: One number exceeds another number by [latex]17[\/latex] and their sum is [latex]31[\/latex]. Find the two numbers.\r\n\r\n[reveal-answer q=\"460075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"460075\"]\r\n\r\nLet [latex]x[\/latex] equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as [latex]x+17[\/latex]. The sum of the two numbers is 31. We usually interpret the word <em>is<\/em> as an equal sign.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{rl}x+\\left(x+17\\right)&amp;=31\\hfill \\\\ 2x+17&amp;=31\\hfill&amp;\\text{Simplify and solve}.\\hfill \\\\ 2x&amp;=14\\hfill \\\\ x&amp;=7\\hfill\\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n<div>The second number would then be [latex]x+17=7+17=24[\/latex]<\/div>\r\n<div><\/div>\r\n<div>The two numbers are [latex]7[\/latex] and [latex]24[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind a linear equation to solve for the following unknown quantities: One number is three more than twice another number. If the sum of the two numbers is [latex]36[\/latex], find the numbers.\r\n[reveal-answer q=\"930268\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930268\"]\r\n\r\n11 and 25[\/hidden-answer]\r\n\r\n[ohm_question height=\"335\"]142770-142775[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall the forms of equations of lines<\/h3>\r\n<strong>Slope-intercept form<\/strong>: [latex]y=mx+b[\/latex], where [latex]x \\text{and } y[\/latex] represent the coordinates of any point on the line, [latex]m[\/latex] represents the slope of the line, and [latex]b[\/latex] represents the initial value, or the y-intercept. You can solve for [latex]b[\/latex] by substituting a known point for [latex]x \\text{ and } y[\/latex] and the slope of the line for [latex]m[\/latex].\r\n\r\nThe <strong>point-slope form<\/strong>, [latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex] simplifies to slope-intercept form when solved for [latex]y[\/latex]. Substitute the coordinates of a point for [latex]{y}_1 \\text{ and } {x}_1[\/latex] and the slope for [latex]m[\/latex] then simplify.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: write a Linear Equation in the form of to Solve a Real-World Application<\/h3>\r\nThere are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05\/min talk-time. Company B charges a monthly service fee of $40 plus $.04\/min talk-time.\r\n<ol>\r\n \t<li>Write a linear equation that models the packages offered by both companies.<\/li>\r\n \t<li>If the average number of minutes used each month is 1,160, which company offers the better plan?<\/li>\r\n \t<li>If the average number of minutes used each month is 420, which company offers the better plan?<\/li>\r\n \t<li>How many minutes of talk-time would yield equal monthly statements from both companies?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"785384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785384\"]\r\n<ol>\r\n \t<li>The model for Company <em>A<\/em> can be written as [latex]A=0.05x+34[\/latex]. This includes the variable cost of [latex]0.05x[\/latex] plus the monthly service charge of $34. Company <em>B<\/em>\u2019s package charges a higher monthly fee of $40, but a lower variable cost of [latex]0.04x[\/latex]. Company <em>B<\/em>\u2019s model can be written as [latex]B=0.04x+40[\/latex].<\/li>\r\n \t<li>If the average number of minutes used each month is 1,160, we have the following:\r\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&amp;=0.05\\left(1,160\\right)+34\\hfill \\\\ \\hfill&amp;=58+34\\hfill \\\\ \\hfill&amp;=92\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&amp;=0.04\\left(1,160\\right)+40\\hfill \\\\ \\hfill&amp;=46.4+40\\hfill \\\\ \\hfill&amp;=86.4\\hfill \\end{array}[\/latex]<\/div>\r\nSo, Company <em>B<\/em> offers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by Company <em>A<\/em> when the average number of minutes used each month is 1,160.<\/li>\r\n \t<li>If the average number of minutes used each month is 420, we have the following:\r\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&amp;=0.05\\left(420\\right)+34\\hfill \\\\ \\hfill&amp;=21+34\\hfill \\\\ \\hfill&amp;=55\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&amp;=0.04\\left(420\\right)+40\\hfill \\\\ \\hfill&amp;=16.8+40\\hfill \\\\ \\hfill&amp;=56.8\\hfill \\end{array}[\/latex]<\/div>\r\nIf the average number of minutes used each month is 420, then Company <em>A <\/em>offers a lower monthly cost of $55 compared to Company <em>B<\/em>\u2019s monthly cost of $56.80.<\/li>\r\n \t<li>To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of [latex]\\left(x,y\\right)[\/latex] coordinates: At what point are both the <em>x-<\/em>value and the <em>y-<\/em>value equal? We can find this point by setting the equations equal to each other and solving for <em>x.<\/em>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}0.05x+34=0.04x+40\\hfill \\\\ 0.01x=6\\hfill \\\\ x=600\\hfill \\end{array}[\/latex]<\/div>\r\nCheck the <em>x-<\/em>value in each equation.\r\n<div style=\"text-align: left;\">[latex]\\begin{array}{l}0.05\\left(600\\right)+34=64\\hfill \\\\ 0.04\\left(600\\right)+40=64\\hfill \\end{array}[\/latex]<\/div>\r\nTherefore, a monthly average of 600 talk-time minutes renders the plans equal.<\/li>\r\n<\/ol>\r\n<ol>\r\n \t<li><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200339\/CNX_CAT_Figure_02_03_002.jpg\" alt=\"Coordinate plane with the x-axis ranging from 0 to 1200 in intervals of 100 and the y-axis ranging from 0 to 90 in intervals of 10. The functions A = 0.05x + 34 and B = 0.04x + 40 are graphed on the same plot\" width=\"731\" height=\"420\" \/>[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\nThe following video shows another example of using linear equations to model and compare two cell phone plans.\r\n\r\nhttps:\/\/youtu.be\/Q5hlC_VPKGM\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nThere are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $0.04\/min talk-time. Company B charges a monthly service fee of $45 plus $0.03\/min talk-time. Use [latex]x[\/latex] for your variable.\r\n\r\na) Write an equation that models the monthly cost for company A.\r\n\r\nb) Write an equation that models the monthly cost for company B.\r\n\r\nC) If the average number of minutes used each month is 1,162, how much is the monthly cost for each company?\r\n\r\n[reveal-answer q=\"981057\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"981057\"]\r\n\r\na) [latex]A=0.04x+34[\/latex]\r\n\r\nb) [latex]B = 0.03x+45[\/latex]\r\n\r\nc) Company A's cost would be $80.48, and Company B's cost would be $79.86\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"415\"]92426[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind a linear equation to model this real-world application: It costs ABC electronics company $2.50 per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of $350 for utilities and $3,300 for salaries. What are the company\u2019s monthly expenses?\r\n[reveal-answer q=\"68149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"68149\"][latex]\r\n\r\nC=2.5x+3,650[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"185\"]93001[\/ohm_question]\r\n\r\n<\/div>\r\nWe can also use a linear equation in one variable to solve a problem with two unknowns by writing an expression for one unknown in terms of the other.\r\n<div class=\"textbox exercises\">\r\n<h3>example: write a linear equation in one variable to model and solve an application<\/h3>\r\nA bag is filled with green and blue marbles. There are 77 marbles in the bag. If there are 17 more green marbles than blue marbles, find the number of green marbles and the number of blue marbles in the bag.\r\n\r\nHow many marbles of each color are in the bag?\r\n\r\n[reveal-answer q=\"470923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"470923\"]\r\n\r\nLet [latex]G[\/latex] represent the number of green marbles in the bag and [latex]B[\/latex] represent the number of blue marbles. We have that [latex]G + B = 77[\/latex]. But we also have that there are 17 more green marbles than blue. That is, the number of green marbles is the same as the number of blue marbles plus 17. We can translate that as [latex]G = B+17[\/latex]. Since we've found a way to express the variable [latex]G[\/latex] in terms of [latex]B[\/latex], we can write one equation in one variable.\r\n\r\n[latex]B+17+B=77[\/latex], which we can solve for [latex]B[\/latex].\r\n\r\n[latex]2B=60[\/latex], so [latex]B=30[\/latex]\r\n\r\nThere are [latex]30[\/latex] Blue marbles in the bag and [latex]30+17 = 47[\/latex] is the number of green marbles.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video for another example of using a linear equation in one variable to solve an application.\r\n\r\nhttps:\/\/youtu.be\/vOA7SGQCpr8\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nA cash register contains only five dollar and ten dollar bills. It contains twice as many five's as ten's and the total amount of money in the cash register is 560 dollars. How many ten's are in the cash register?\r\n\r\n[reveal-answer q=\"315017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"315017\"]There are 28 ten dollar bills in the cash register.[\/hidden-answer]\r\n\r\n[ohm_question height=\"200\"]13829[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write a linear equation to express the relationship between unknown quantities.<\/li>\n<li>Write a linear equation to model a real-world situation.<\/li>\n<li>Write a linear equation in one variable to solve problems with two unknowns.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall expressions and equations<\/h3>\n<p>Recall that a mathematical\u00a0<strong>expression<\/strong>\u00a0consists of <strong>terms<\/strong> connected by addition or subtraction, each term of which consists of <strong>variables<\/strong> and numbers connected by multiplication or division. A number that multiplies a variable, such as the [latex]2[\/latex] in the term [latex]2x[\/latex] is called the\u00a0<strong>coefficient\u00a0<\/strong>of the variable.<\/p>\n<p>An\u00a0<strong>equation<\/strong> is a mathematical statement of the equivalency of two expressions, one on either side of an equal sign.<\/p>\n<p><strong>Expressions<\/strong> may be combined or simplified by rearranging or doing operations on the terms. <strong>Equations<\/strong> may be solved for the value or values of the variable that make the statement of equivalency true.<\/p>\n<\/div>\n<p>Consider a car rental agency that charges $150 per week plus $0.10 per mile driven to rent a compact car. We can use these quantities to write an equation that models the cost of renting the car for a week [latex]C[\/latex] given a certain number of miles [latex]x[\/latex] driven.<\/p>\n<div style=\"text-align: center;\">[latex]C=150+0.10x[\/latex]<\/div>\n<p>To set up a linear equation that models a real-world situation, we must first determine the known quantities and define the unknown quantity as a variable. Then, we interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10 per mile, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write [latex]0.10x[\/latex] to model the portion of the weekly cost generated by miles driven. This expression represents a variable cost because it changes according to the number of miles driven.<\/p>\n<p>If a quantity is independent of a variable, we usually just add or subtract it according to the problem. In the car rental example, there is a flat fee of $150 to rent the car, independent of the number of miles driven. In applications involving costs, amounts such as this flat fee that do not change are often called fixed costs.<\/p>\n<p>When dealing with real-world applications, there are certain expressions that we can translate directly into math. The table below lists some common verbal expressions and their equivalent mathematical expressions.<\/p>\n<table summary=\"A table with 8 rows and 2 columns. The entries in the first row are: Verbal and Translation to math operations. The entries in the second row are: One number exceeds another by a and x, x+a. The entries in the third row are: Twice a number and 2x. The entries in the fourth row are: One number is a more than another number and x, x plus a. The entries in the fifth row are: One number is a less than twice another number and x,2 times x minus a. The entries in the sixth row are: The product of a number and a, decreased by b and a times x minus b. The entries in the seventh row are: The quotient of a number and the number plus a is three times the number and x divided by the quantity x plus a equals three times x. The entries in the eighth row are: The product of three times a number and the number decreased by b is c and three times x times the quantity x minus b equals c.\">\n<thead>\n<tr>\n<th>Verbal<\/th>\n<th>Translation to Math Operations<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>One number exceeds another by <em>a<\/em><\/td>\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Twice a number<\/td>\n<td>[latex]2x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>One number is <em>a <\/em>more than another number<\/td>\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>One number is <em>a <\/em>less than twice another number<\/td>\n<td>[latex]x,2x-a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product of a number and <em>a<\/em>, decreased by <em>b<\/em><\/td>\n<td>[latex]ax-b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The quotient of a number and the number plus <em>a <\/em>is three times the number<\/td>\n<td>[latex]\\Large\\frac{x}{x+a}\\normalsize =3x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product of three times a number and the number decreased by <em>b <\/em>is <em>c<\/em><\/td>\n<td>[latex]3x\\left(x-b\\right)=c[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the car-rental example above, we identified the unknown quantity (number of miles driven), and assigned it the variable [latex]x[\/latex]. Then we identified known quantities, and translated the words in the situation into relationships between the known and unknown quantities to write an equation that models the situation. We could use the model to answer questions about the situation such as finding the total cost for a week if 500 miles were driven or how many miles could be driven in a week on a budget of $375.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a real-world situation, write a linear equation to model it<\/h3>\n<ol>\n<li>Identify known and unknown quantities.<\/li>\n<li>Assign a variable to represent the unknown quantity.<\/li>\n<li>If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.<\/li>\n<li>Write an equation interpreting the words in the problem as mathematical operations.<\/li>\n<li>Solve the equation, check to be sure your answer is reasonable, and give the answer using the language and units of the original situation.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find a linear equation to solve for the following unknown quantities: One number exceeds another number by [latex]17[\/latex] and their sum is [latex]31[\/latex]. Find the two numbers.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q460075\">Show Solution<\/span><\/p>\n<div id=\"q460075\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]x[\/latex] equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as [latex]x+17[\/latex]. The sum of the two numbers is 31. We usually interpret the word <em>is<\/em> as an equal sign.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rl}x+\\left(x+17\\right)&=31\\hfill \\\\ 2x+17&=31\\hfill&\\text{Simplify and solve}.\\hfill \\\\ 2x&=14\\hfill \\\\ x&=7\\hfill\\end{array}[\/latex]<\/div>\n<div><\/div>\n<div>The second number would then be [latex]x+17=7+17=24[\/latex]<\/div>\n<div><\/div>\n<div>The two numbers are [latex]7[\/latex] and [latex]24[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find a linear equation to solve for the following unknown quantities: One number is three more than twice another number. If the sum of the two numbers is [latex]36[\/latex], find the numbers.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q930268\">Show Solution<\/span><\/p>\n<div id=\"q930268\" class=\"hidden-answer\" style=\"display: none\">\n<p>11 and 25<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm142770\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142770-142775&theme=oea&iframe_resize_id=ohm142770&show_question_numbers\" width=\"100%\" height=\"335\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall the forms of equations of lines<\/h3>\n<p><strong>Slope-intercept form<\/strong>: [latex]y=mx+b[\/latex], where [latex]x \\text{and } y[\/latex] represent the coordinates of any point on the line, [latex]m[\/latex] represents the slope of the line, and [latex]b[\/latex] represents the initial value, or the y-intercept. You can solve for [latex]b[\/latex] by substituting a known point for [latex]x \\text{ and } y[\/latex] and the slope of the line for [latex]m[\/latex].<\/p>\n<p>The <strong>point-slope form<\/strong>, [latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex] simplifies to slope-intercept form when solved for [latex]y[\/latex]. Substitute the coordinates of a point for [latex]{y}_1 \\text{ and } {x}_1[\/latex] and the slope for [latex]m[\/latex] then simplify.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: write a Linear Equation in the form of to Solve a Real-World Application<\/h3>\n<p>There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05\/min talk-time. Company B charges a monthly service fee of $40 plus $.04\/min talk-time.<\/p>\n<ol>\n<li>Write a linear equation that models the packages offered by both companies.<\/li>\n<li>If the average number of minutes used each month is 1,160, which company offers the better plan?<\/li>\n<li>If the average number of minutes used each month is 420, which company offers the better plan?<\/li>\n<li>How many minutes of talk-time would yield equal monthly statements from both companies?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785384\">Show Solution<\/span><\/p>\n<div id=\"q785384\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The model for Company <em>A<\/em> can be written as [latex]A=0.05x+34[\/latex]. This includes the variable cost of [latex]0.05x[\/latex] plus the monthly service charge of $34. Company <em>B<\/em>\u2019s package charges a higher monthly fee of $40, but a lower variable cost of [latex]0.04x[\/latex]. Company <em>B<\/em>\u2019s model can be written as [latex]B=0.04x+40[\/latex].<\/li>\n<li>If the average number of minutes used each month is 1,160, we have the following:\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&=0.05\\left(1,160\\right)+34\\hfill \\\\ \\hfill&=58+34\\hfill \\\\ \\hfill&=92\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&=0.04\\left(1,160\\right)+40\\hfill \\\\ \\hfill&=46.4+40\\hfill \\\\ \\hfill&=86.4\\hfill \\end{array}[\/latex]<\/div>\n<p>So, Company <em>B<\/em> offers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by Company <em>A<\/em> when the average number of minutes used each month is 1,160.<\/li>\n<li>If the average number of minutes used each month is 420, we have the following:\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&=0.05\\left(420\\right)+34\\hfill \\\\ \\hfill&=21+34\\hfill \\\\ \\hfill&=55\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&=0.04\\left(420\\right)+40\\hfill \\\\ \\hfill&=16.8+40\\hfill \\\\ \\hfill&=56.8\\hfill \\end{array}[\/latex]<\/div>\n<p>If the average number of minutes used each month is 420, then Company <em>A <\/em>offers a lower monthly cost of $55 compared to Company <em>B<\/em>\u2019s monthly cost of $56.80.<\/li>\n<li>To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of [latex]\\left(x,y\\right)[\/latex] coordinates: At what point are both the <em>x-<\/em>value and the <em>y-<\/em>value equal? We can find this point by setting the equations equal to each other and solving for <em>x.<\/em>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}0.05x+34=0.04x+40\\hfill \\\\ 0.01x=6\\hfill \\\\ x=600\\hfill \\end{array}[\/latex]<\/div>\n<p>Check the <em>x-<\/em>value in each equation.<\/p>\n<div style=\"text-align: left;\">[latex]\\begin{array}{l}0.05\\left(600\\right)+34=64\\hfill \\\\ 0.04\\left(600\\right)+40=64\\hfill \\end{array}[\/latex]<\/div>\n<p>Therefore, a monthly average of 600 talk-time minutes renders the plans equal.<\/li>\n<\/ol>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200339\/CNX_CAT_Figure_02_03_002.jpg\" alt=\"Coordinate plane with the x-axis ranging from 0 to 1200 in intervals of 100 and the y-axis ranging from 0 to 90 in intervals of 10. The functions A = 0.05x + 34 and B = 0.04x + 40 are graphed on the same plot\" width=\"731\" height=\"420\" \/><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p>The following video shows another example of using linear equations to model and compare two cell phone plans.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Write Linear Equations to Model and Compare Cell Phone Plans with Data Usage\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Q5hlC_VPKGM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $0.04\/min talk-time. Company B charges a monthly service fee of $45 plus $0.03\/min talk-time. Use [latex]x[\/latex] for your variable.<\/p>\n<p>a) Write an equation that models the monthly cost for company A.<\/p>\n<p>b) Write an equation that models the monthly cost for company B.<\/p>\n<p>C) If the average number of minutes used each month is 1,162, how much is the monthly cost for each company?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q981057\">Show Solution<\/span><\/p>\n<div id=\"q981057\" class=\"hidden-answer\" style=\"display: none\">\n<p>a) [latex]A=0.04x+34[\/latex]<\/p>\n<p>b) [latex]B = 0.03x+45[\/latex]<\/p>\n<p>c) Company A&#8217;s cost would be $80.48, and Company B&#8217;s cost would be $79.86<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm92426\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92426&theme=oea&iframe_resize_id=ohm92426&show_question_numbers\" width=\"100%\" height=\"415\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find a linear equation to model this real-world application: It costs ABC electronics company $2.50 per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of $350 for utilities and $3,300 for salaries. What are the company\u2019s monthly expenses?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q68149\">Show Solution<\/span><\/p>\n<div id=\"q68149\" class=\"hidden-answer\" style=\"display: none\">[latex]C=2.5x+3,650[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm93001\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93001&theme=oea&iframe_resize_id=ohm93001&show_question_numbers\" width=\"100%\" height=\"185\"><\/iframe><\/p>\n<\/div>\n<p>We can also use a linear equation in one variable to solve a problem with two unknowns by writing an expression for one unknown in terms of the other.<\/p>\n<div class=\"textbox exercises\">\n<h3>example: write a linear equation in one variable to model and solve an application<\/h3>\n<p>A bag is filled with green and blue marbles. There are 77 marbles in the bag. If there are 17 more green marbles than blue marbles, find the number of green marbles and the number of blue marbles in the bag.<\/p>\n<p>How many marbles of each color are in the bag?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q470923\">Show Solution<\/span><\/p>\n<div id=\"q470923\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]G[\/latex] represent the number of green marbles in the bag and [latex]B[\/latex] represent the number of blue marbles. We have that [latex]G + B = 77[\/latex]. But we also have that there are 17 more green marbles than blue. That is, the number of green marbles is the same as the number of blue marbles plus 17. We can translate that as [latex]G = B+17[\/latex]. Since we&#8217;ve found a way to express the variable [latex]G[\/latex] in terms of [latex]B[\/latex], we can write one equation in one variable.<\/p>\n<p>[latex]B+17+B=77[\/latex], which we can solve for [latex]B[\/latex].<\/p>\n<p>[latex]2B=60[\/latex], so [latex]B=30[\/latex]<\/p>\n<p>There are [latex]30[\/latex] Blue marbles in the bag and [latex]30+17 = 47[\/latex] is the number of green marbles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for another example of using a linear equation in one variable to solve an application.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solve a Coin Problem Using an Equation in One Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vOA7SGQCpr8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>A cash register contains only five dollar and ten dollar bills. It contains twice as many five&#8217;s as ten&#8217;s and the total amount of money in the cash register is 560 dollars. How many ten&#8217;s are in the cash register?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q315017\">Show Solution<\/span><\/p>\n<div id=\"q315017\" class=\"hidden-answer\" style=\"display: none\">There are 28 ten dollar bills in the cash register.<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm13829\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=13829&theme=oea&iframe_resize_id=ohm13829&show_question_numbers\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\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-99\">\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\/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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 142770, 142775. <strong>Authored by<\/strong>: Alyson Day. <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 13829. <strong>Authored by<\/strong>: James Sousa. <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 92426, 93001. <strong>Authored by<\/strong>: Michael Jenck. <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>Writing and Solving Linear Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Q5hlC_VPKGM\">https:\/\/youtu.be\/Q5hlC_VPKGM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solve a Coin Problem Using an Equation in One Variable. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vOA7SGQCpr8\">https:\/\/youtu.be\/vOA7SGQCpr8<\/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\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Learn Desmos: Change Graph Settings. <strong>Authored by<\/strong>: Desmos. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/En_PkyA-4_4\">https:\/\/youtu.be\/En_PkyA-4_4<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube Licesnse<\/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":395986,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"copyrighted_video\",\"description\":\"Learn Desmos: Change Graph 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