Section 2.1 Equations and Their Solutions 95 Solution 3b. What is unknown? The number is unknown. Let n represent the number. What is known? The difference of three times a number and 6 is the same as the quotient of the number and 9. The phrase “is the same as” represents the equals sign. “The difference of three times a number and 6” represents the left side of the equation. “The quotient of the number and 9” represents the right side of the equation. The difference of three times a number and 6 is the same as the quotient of the number and 9 3n - 6 = n 9 So, the equation used to solve this problem is 3n - 6 = n 9 . Student Check 3 For each problem, define a variable and write an equation that can be used to solve the problem. a. Three times the sum of a number and five is the same as four times the number. b. The sum of twice a number and one is equal to two more than the number. Representing Applications Algebraically Setting up equations for application problems is very similar to the previous objective. The main difference is that we may have to rely on key words in the problem that define the relationships implicitly. For instance, we might be told the perimeter of an object or that two angles are complementary. In these cases, we have to know how to calculate the perimeter of the object or what it means for two angles to be complementary to set up the equation. Objective 4 Examples For each problem, define a variable and write an equation that can be used to solve the problem. 4a. One angle is 30° more than another angle. The angles are complementary. Find the measure of each angle. Solution 4a. What is unknown? The measure of the two angles is unknown. Let a represent the measure of one of the angles. The other angle is 30° more, so a + 30 represents the measure of the other angle. What is known? The angles are complementary, so their sum is 90°. The phrase “is” represents the equals sign. “Their sum” represents the left side of the equation. “90°” represents the right side of the equation. Their sum is 90° a + a + 30 = 90° So, the equation used to solve this problem is a + a + 30 = 90°. 4b. From June 2009 to June 2010, Oprah Winfrey and James Cameron, director of the movies Avatar and Titantic, were the two highest paid celebrities. They earned a combined total of $525 million. Winfrey earned $105 million more than Cameron. How much did each of them earn (in millions)? (Source: http:// www.forbes.com/lists/) Objective 4 ▶ Set up equations for application problems. a 30 a
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