Equations by Addition or Subtraction

Transcription

1 Lesson 8.4 Objectives Solve systems of linear equations by addition or subtraction. Solving Systems of Equations by Addition or Subtraction Nancy and Jim are team leaders at a computer chip manufacturing company. The production supervisor needs to report the number of computer chips each team made on Friday. The supervisor knows the total number of chips produced by both teams is 130. Nancy s team made 10 more chips than Jim s team. How many computer chips did each team make? Solve by Addition Both the graphing and substitution methods can be used to solve systems of equations. However, you can also use another method based on an alternate form of the Addition Property of Equality. Addition Property of Equality (Alternate Form) For all numbers a, b, c, and d, if a b and c d, then a c b d. This property allows you to form a new equation from two existing equations. For example, suppose you have the following system of equations. a b c d If you add the left sides and then the right sides, a new equation results. a b c d a c b d The reason for using the Addition Property of Equality to form a new equation is that in certain cases the new equation will contain variables that are opposites. Because opposites sum to zero, the new equation will contain only one variable, which you can solve using the methods you previously learned. 8.4 Solving Systems of Equations by Addition or Subtraction 463

2 Example 1 Solving by Addition Reread the opening paragraph of this lesson. How many computer chips did Nancy s team and Jim s team make? Solution Let x represent the number of chips made by Nancy s team. Let y represent the number of chips made by Jim s team. The following system models the information known by the supervisor. x y 130 x y 10 Use the Addition Property of Equality to add both sides of the equations. (x y) (x y) Notice that y and 2y are opposites, and their sum is zero. Thus the y variable is eliminated, leaving an equation with only the single variable x. (x x) x 140 x 70 Substituting this value for x in either of the original equations results in y 60. The ordered pair (70, 60) is the solution for the system of equations. Nancy s team made 70 computer chips and Jim s team made 60 computer chips on Friday. Ongoing Assessment Solve this system of equations using the alternate form of the Addition Property of Equality. (60, 40) x y 100 x y 20 Subtraction Property of Equality You can also use an alternate form of the Subtraction Property of Equality to solve systems of linear equations. Subtraction Property of Equality (Alternate Form) For all numbers a, b, c, and d, If a b and c d, then a c b d. 464 Chapter 8 Systems of Equations

3 Example 2 Solving by Subtraction Subtraction can also solve the computer chip production system of equations. x y 130 x y 10 This time subtract the expressions on each side of the equal signs. Notice in the following steps how the x-variable is eliminated. x y 130 x y 130 (x y) 10 x y 10 2y 120 y 60 The result of replacing y with 60 in either original equation gives x 70. Example 3 Solving by Subtraction Solve this system of equations. 2x 2y 10 3x 2y 14 Solution Use the alternate form of the Subtraction Property of Equality to subtract the expressions on the left side and the expressions on the right side of the equal sign. 2x 2y 10 2x 2y 10 (3x 2y) 14 3x 2y 14 x 0y 4 x 4 x 4 To solve for y, substitute 4 for x in the first equation. 2(4) 2y y 10 2y 2 y 1 The solution to the system of equations is (4, 1). Check this solution in both equations. 8.4 Solving Systems of Equations by Addition or Subtraction 465

4 Ongoing Assessment Use the alternate form of the Subtraction Property of Equality to solve the following system of equations. 3 2, 1 2x y 2 2x 3y 0 Activity Solving a System of Equations by Subtracting On Monday, two automobile manufacturing plants produced a total of 80 vehicles. Nancy s plant made four fewer than twice the number produced by Jim s plant. Let x represent the number of vehicles made by Nancy s plant. Let y represent the number of vehicles made by Jim s plant. 1 What equation models the total number of vehicles made by the plants? x y 80 2 Write an equation that models the number of vehicles made by Nancy s plant. x 4 2y 3 Write the equations from Steps 1 and 2 in the standard form ax by c. x y 80; x 2y 4 4 Solve this system of equations using the alternate form of the Subtraction Property of Equality. x 52; y 28 5 How many vehicles did each plant produce on Monday? Nancy s plant: 52; Jim s plant: 28 You can use the addition or subtraction method to solve a linear system. Write each equation in standard form, ax by c. Add or subtract the terms on the same side of the equal signs of the two equations to eliminate one of the variables from the system. Then solve for the other variable. 466 Chapter 8 Systems of Equations

5 Cultural Connection Chinese mathematicians solved systems of equations even before they invented the abacus. These mathematicians used colored rods to represent numbers in the system. For example, let green rods represent positive numbers and red rods represent negative numbers. Explain how the diagram is used to solve the system for x. see margin 3x y 3 2x y 1 Use the Chinese method to solve for y. What is the solution to the system of equations? (2, 23) Lesson Assessment Think and Discuss 1. When is the addition method the best method for solving a system of equations in standard form? When is the subtraction method best? 2. How do you write an equation in standard form? 3. Why should the two equations be written in standard form before a system of equations is solved? 4. Explain how to solve the system of equations below. 2x 5 3y 2 5 3y 1 4x Solving Systems of Equations by Addition or Subtraction 467

6 Practice and Problem Solving Solve by addition or subtraction. 5. 2y 2x y 3x x 4y 4 2y 5x 9 ( 2, 1 2 ) 3y 3x 0 4y 6x 2. (0, 0). inconsistent 8. x 3y 6 9. x y x y 6.5 x y 6 2x y 1 x y 2.5 (3, 3). (1, 1). (2, 0.5) 11. 2x y x y x y 2 2x 3y 14 x y 4 2x y 1 ( 2, 6). (3, 1). (3, 5) 14. x y x 5y x 33 5y 3y x 17 2x 6y 10 x 7y 45 ( 8, 3). (7, 4). ( 3, 6) 17. 2x 4y y 4x x 5y 14 3x 4y 3 4x 2y 18 3x 6y 30 ( 7, 6). (7, 5). ( 2, 4) Write a system of equations for each situation. Solve and check your answers. 20. Because of limited storage space at the job site, a bricklayer must schedule two deliveries. The two deliveries must contain a total of 23 pallets of bricks. One delivery must have three more pallets than the other. How many pallets of bricks will be in each delivery? 13; In a surveyor s drawing using a coordinate system, one side of an angle lies along the line x y 3, and the other side lies along the line 2x y 9. Find the coordinates of the vertex of the angle. (6, 3) 22. To place a newspaper ad, you must pay a fixed (or flat) rate for the first ten words and a fixed charge for each additional word. The cost to place a 17-word ad is $ The cost for a 21-word ad is $ What is the flat rate, and what is the fixed charge for each additional word? $10; $0.65 Mixed Review 468 Chapter 8 Systems of Equations 23. Let g(x) 1 x. Let h(x) 2x. a. Find g(5). 1 b. Find h(2). 4 c. Find g(0.5) d. Find h(2.5). 5 e. Find g(h(x)). 2 f. Find h(g(x)). 2 x x g. Graph g in intervals of 0.5 for the domain from 0.5 through 5. see margin h. What is the range for the domain given in part g? 0.2 y 2 i. What happens to g(x) as x becomes very large? g(x) becomes very small.