1. a. By what nonzero constant would the second equation be multiplied to eliminate the xvariable from the system of equations?

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1 Section 4.3 Practice Exercises Study Skills Exercise Instructors differ in what they emphasize on tests. For example, test material may come from the textbook, notes, handouts, or homework. What does your instructor emphasize? Vocabulary and Key Concepts 1. a. By what nonzero constant would the second equation be multiplied to eliminate the xvariable from the system of equations? b. By what nonzero constant would the first equation be multiplied to eliminate the y variable from the system of equations? Review Exercises For Exercises 2 4, use the slope-intercept form of the lines to determine the number of solutions for the system of equations Concept 1: The Addition Method For Exercises 5 16, solve the system by the addition method. (SeeExamples 1 3.)

2 Concept 2: Solving Inconsistent Systems and Systems of Dependent Equations For Exercises 17 24, solve the system. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (See Examples 4and 5.)

3 Page 262 Mixed Exercises 25. Describe a situation in which you would prefer to use the substitution method over the addition method. 26. If you used the addition method to solve the given system, would it be easier to eliminate the x or yvariable? Explain. For Exercises 27 53, solve by using either the addition method or the substitution method. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent

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6 Section 4.6 Practice Exercises Study Skills Exercise Look back over your notes for this chapter. Have you highlighted the important topics? Have you underlined the key terms? Have you indicated the places where you are having trouble? If you find that you have problems with a particular topic, write a question that you can ask your instructor either in class or in the office. Vocabulary and Key Concepts 1. a. An equation written in the form Ax + By + Cz = D where A, B, and C are not all zero is called a equation in three variables. b. Solutions to a linear equation in three variables are of the form (x, y, z) and are called. Review Exercises For Exercises 2 4, solve the systems by using two methods: (a) the substitution method and (b) the addition method Marge can ride her bike 24 mi in hr riding with the wind. Riding against the wind she can ride 24 mi in 2 hr. Find the speed at which Marge can ride in still air and the speed of the wind. Animation: Introduces systems of linear equations in three variables Animation: Introduces systems of linear equations in three variables (Please note that browsers Safari and Google Chrome do not support all animations and therefore users may not be able to view the animations on these browsers.) Page 293 Concept 1: Solutions to Systems of Linear Equations in Three Variables 6. How many solutions are possible when solving a system of three equations with three variables? 7. Which of the following points are solutions to the system? (2, 1, 7), (3, 10, 6), (4, 0, 2) 8. Which of the following points are solutions to the system? (1, 1, 3), (0, 0, 4), (4, 2, 1) 9. Which of the following points are solutions to the system? (12, 2, 2), (4, 2, 1), (1, 1, 1)

7 10. Which of the following points are solutions to the system? (0, 4, 3), (3, 6, 10), (3, 3, 1) Concept 2: Solving Systems of Linear Equations in Three Variables For Exercises 11 22, solve the system of equations. (See Example 1.) Exercise: Solve Systems of Linear Equations in Three Variables PDF Transcript for Exercise: Solve Systems of Linear Equations in Three Variables

8 Exercise: Solving Systems of Linear Equations in Three Variables PDF Transcript for Exercise: Solving Systems of Linear Equations in Three Variables Concept 3: Applications of Linear Equations in Three Variables 23. A triangle has one angle that measures 5 more than twice the smallest angle, and the third angle measures 11 less than 3 times the measure of the smallest angle. Find the measures of the three angles. (See Example 2.) Exercise: Application of Linear Equations in Three Variables PDF Transcript for Exercise: Application of Linear Equations in Three Variables 24. The largest angle of a triangle measures 4 less than 5 times the measure of the smallest angle. The middle angle measures twice that of the smallest angle. Find the measures of the three angles. 25. The perimeter of a triangle is 55 cm. The measure of the shortest side is 8 cm less than the middle side. The measure of the longest side is 1 cm less than the sum of the other two sides. Find the lengths of the sides. 26. The perimeter of a triangle is 5 ft. The longest side of the triangle measures 20 in. more than the shortest side. The middle side is 3 times the measure of the shortest side. Find the lengths of the three sides in inches. Page Sean kept track of his fiber intake from three sources for 3 weeks. The first week he had 3 servings of a fiber supplement, 1 serving of oatmeal, and 4 servings of cereal, which totaled 19 g of fiber. The second week he had 2 servings of the fiber supplement, 4 servings of oatmeal, and 2 servings of cereal totaling 25 g. The third week he had 5 servings of the fiber supplement, 3 servings of oatmeal, and 2 servings of cereal for a total of 30 g. Find the amount of fiber in one serving of each of the following: the fiber supplement, the oatmeal, and the cereal. (See Example 3.) 28. Natalie kept track of her calcium intake from three sources for 3 days. The first day she had 1 glass of milk, 1 serving of ice cream, and 1 calcium supplement in pill form which totaled 1180 mg of calcium. The second day she had 2 glasses of milk, 1 serving of ice cream, and 1 calcium supplement totaling 1680 mg. The third day she had 1 glass of milk, 2 servings of ice cream, and 1 calcium supplement for a total of 1260 mg. Find the amount of calcium in one glass of milk, in one serving of ic e cream, and in one calcium supplement. 29. Goofie Golf has 18 holes that are par 3, par 4, or par 5. Most of the holes are par 4. In fact, there are 3 times as many par 4s as par 3s. There are 3 more par 5s than par 3s. How many of each type are there? 30. Combining peanuts, pecans, and cashews makes a party mixture of nuts. If the amount of peanuts equals the amount of pecans and cashews combined, and if there are twice as many cashews as pecans, how many ounces of each nut is used to make 48 oz of party mixture?

9 Exercise: Applications of Linear Equations in Three Variables PDF Transcript for Exercise: Applications of Linear Equations in Three Variables 31. Souvenir hats, T-shirts, and jackets are sold at a rock concert. Three hats, two T-shirts, and one jacket cost $140. Two hats, two T-shirts, and two jackets cost $170. One hat, three T-shirts, and two jackets cost $180. Find the prices of the individual items. 32. Annie and Maria traveled overseas for 7 days and stayed in three different hotels in three different cities: Stockholm, Sweden; Oslo, Norway; and Paris, France. The total bill for all seven nights (not including tax) was $1040. The total tax was $106. The nightly cost (excluding tax) to stay at the hotel in Paris was $80 more than the nightly cost (excluding tax) to stay in Oslo. Find the cost per night for each hotel excluding tax. City Number of Nights Cost/Night ($) Tax Rate Paris, France 1 x 8% Stockholm, Sweden 4 y 11% Oslo, Norway 2 z 10% 33. Walter had $25,000 to invest. He split the money into three types of investment: small caps earning 6%, global market investments earning 10%, and a balanced fund earning 9%. He put twice as much money in the global account as he did in the balanced fund. If his earnings for the first year totaled $2160, how much did he invest in each account? 34. Raeann deposited $8000 into three accounts at her credit union: a checking account that pays 1.2% interest, a savings account that pays 2.5% interest, and a money market account that pays 3% interest. If she put three times more money in the 3% account than she did in the 1.2% account, and her total interest for 1 year was $202, how much did she deposit into each account? Page 295 Concept 4: Solving Inconsistent Systems and Systems of Dependent Equations (Mixed Exercises) For Exercises 35 46, solve the system. For systems that do not have one unique solution, state whether the system is inconsistent or the equations are dependent. (See Examples 1, 4, and 5.) Exercise: Solving Dependent and Inconsistent Systems of Linear Equations in Three Variables PDF Transcript for Exercise: Solving Dependent and Inconsistent Systems of Linear Equations in Three Variables

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