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Applications Using Linear Equations This section will discuss several types of applications. No matter how simple or complex the application is, the following steps will help to translate and solve the problem. 1. 2. 3. 4. 5. Read through the entire problem Organize the information Write the equation Solve the equation Check the answer Direct Translation With this type of application, you will translate word to symbol. Look for key words and phrases. The following table gives some common translations. Word or Phrase is the same as sum more than difference less than product of Symbol = = + + × × Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 1: The sum of four and a number is ten. The sum of four and a number is ten. 4 + x = 10 -4 -4 x= 6 Subtract 4 from both sides Check: The sum of 4 and 6 is 10. Or 4+6=10. Example 2: If 28 less than five times a number is 232. What is the number? If 28 less than five times a number is 232. 5x – 28 = 232 + 28 + 28 Add 28 to both sides 5x = 260 5 5 Divide both sides by 5 x = 52 Check: 28 less than 5(52) is 232. Or 5(52) – 28 = 232 260 – 28 = 232 232 = 232 Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Consecutive Integers Consecutive integers are numbers that come one after the other, such as 3, 4, 5. To get from one number to the next, we only have to add 1, so if the first number is x, the next number is x+1. Consecutive odd or even integers are numbers that are spaced apart by two, such as 2, 4, 6 or 3, 5, 7. If the first number is x, the next number is x+2. Example 3: The sum of two consecutive integers is 93. What are the integers? The first number is x The second number is x+1 x + x + 1 = 93 2x + 1 = 93 Combine like terms x + x - 1 - 1 Subtract 1 from both sides 2x = 92 2 2 Divide both sides by 2 x = 46 The first number x = 46 The second number x+1 = 46+1 = 47 Check: 47 comes after 46. Also 46+47=93. Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 4: The sum of three consecutive even integers is 246. What are the integers? The first number is x The second number is x+2 The third number is x+4 x + x + 2 + x + 4 = 246 3x + 6 = 246 Combine x + x + x and 2 + 4 -6 - 6 Subtract 6 from both sides 3x = 240 3 3 Divide both sides by 3 x = 80 The first number x = 80 The second number x+2 = 80+2 = 82 The third number x+4 = 80+4 = 84 Check: 80, 82, and 84 are consecutive even numbers. Also 80 + 82 + 84 = 246. Perimeter of a Shape For applications involving perimeter (the distance around an object), drawing a picture is helpful. Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 5: The perimeter of a rectangle is 44 feet. The length is 5 feet less than twice the width. Find the dimensions. 2w – 5 w There are two widths of the same size, w+w or 2w There are two lengths of the same size, (2w-5)+(2w-5) or 2(2w-5) 2w + 2(2w – 5) = 44 2w + 4w – 10 = 44 6w – 10 = 44 + 10 + 10 6w = 54 6 6 w = 9 Distribute the 2 Combine the like terms 2w + 4w Add 10 to both sides Divide both sides by 6 The width w = 9 ft The length 2w-5 = 2(9)-5 = 13 ft Check: Length + Length + Width + Width = 9 + 9 + 13 + 13 = 44 Or 2L + 2W = 2(9) + 2(13) = 18 + 26 = 44 Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 6: If one side of a triangle is twice as long as the shortest side, and the third side is 3 times as long as the shortest side, find the dimensions of a triangle whose perimeter is 72 meters. 2x x 3x 2x + 3x + x = 72 6x = 72 6 6 x = 12 Combine like terms 2x + 3x + x Divide both sides by 6 The shortest side x = 12 m The side twice as long as the shortest 2x = 2(12) = 24 m The side 3 times as long as the shortest 3x = 3(12) = 36 m Check: 2(12) + 3(12) + 12 = 24 + 36 + 12 = 72 Basic “Real Life” Applications The following are examples of a type of application that is very common. They are similar to the above examples but may seem slightly different. When organizing these problems, it is important to clearly identify the variable. Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 7: A sofa and loveseat together costs $444. The cost of the sofa is double the cost of the loveseat. How much do they cost? The love seat is x The sofa is double the love seat, so 2x x + 2x = 444 3x = 444 3 3 x = 148 Combine like terms x + 2x Divide both sides by 3 The love seat x = $148 The sofa 2x = 2(148) = $296 Check: $148 + $296 = $444 Example 8: An eight foot board is cut into two pieces. One piece is 2 feet longer than the other piece. How long are the pieces? x = 8 feet x+x+2=8 2x + 2 = 8 -2 -2 2x =6 2 2 x =3 Combine like terms x + x Subtract 2 from both sides Divide both sides by 2 The short piece x = 3 ft The long piece x + 2 = 3 + 2 = 5 ft Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Basic Percent Problems There are three basic types of percent problems: 1. Find a given percent of a given number. For example: find 25% of 640. 2. Find a percent given two numbers. For example: 15 is what percent of 50? 3. Find a number that is a given percent of another number. For example: 10% of what number is 12? Example 9: What number is 25% of 640? Translate the words into an equation. What number is 25% of 640? x = 25% × 640 x = .25 × 640 x = 160 Change 25% to the decimal .25 160 is 25% of 640 Example 10: 15 is what percent of 50? Translate the words into an equation. 15 is what percent of 50? 15 = x × 50 15 = 50x 50 50 Divide both sides by 50 x = 0.30 = 30% Change 0.30 to the percent 30% 15 is 30% of 50 Modified from Prealgebra Textbook, by the Department of Mathematics, College of the Redwoods, CCBY 2009. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by-nc-sa/3.0) Example 11: 10% of what number is 12? Translate the words into an equation. 10% of what number is 12? 10% × x = 12 .10x = 12 Change 10% to the decimal .10 (100).10x = (100)12 Multiply by 100 to clear the decimal 10x = 1200 10 10 Divide both sides by 10 x = 120 10% of 120 is 12 Modified from Prealgebra Textbook, by the Department of Mathematics, College of the Redwoods, CCBY 2009. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by-nc-sa/3.0)