6-3 Solving Systems by Elimination
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- Neil Boone
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1 Warm Up Simplify each expression. 1. 3x + 2y 5x 2y 2. 5(x y) + 2x + 5y 3. 4y + 6x 3(y + 2x) 4. 2y 4x 2(4y 2x) Write the least common multiple and and and and 5
2 Learning Goals 1. The student is able to solve systems of equations using elimination (addition) 2. The student is able to solve systems of equations using elimination(subtraction) 3. The student is able to solve systems of equations using elimination(multiplication) 4. The student is able to set up a system of equations and solve an application problem
3 Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two equations in the system together. Remember that an equation stays balanced if you add equal amounts to both sides. So, if 5x + 2y = 1, you can add 5x + 2y to one side of an equation and 1 to the other side and the balance is maintained.
4 Since 2y and 2y have opposite coefficients, the y- term is eliminated. The result is one equation that has only one variable: 6x = 18. When you use the elimination method to solve a system of linear equations, align all like terms in the equations. Then determine whether any like terms can be eliminated because they have opposite coefficients.
5 Solving Systems of Equations by Elimination Step 1 Step 2 Step 3 Step 4 Write the system so that like terms are aligned. Eliminate one of the variables and solve for the other variable. Substitute the value of the variable into one of the original equations and solve for the other variable. Write the answers from Steps 2 and 3 as an ordered pair, (x, y), and check.
6 Later in this lesson you will learn how to multiply one or more equations by a number in order to produce opposites that can be eliminated.
7 A. Example 1: Elimination Using Addition Solve by elimination. " 3x 4y =10 # $ x + 4y = 2
8 B. Example 1: Elimination Using Addition Solve by elimination. # y + 3x = 2 $ % 2y 3x =14
9 C. Example 1: Elimination Using Addition Solve by elimination. # x 2y = 19 $ % 5x + 2y =1
10 Example 1: Elimination Using Addition Solve by elimination. " x + y = 5 # $ x 5y = 9 a. b. " x + y =12 # $ x y = 2 c. " 2x + 5y = 24 # $ 3x 5y =14
11 A. Example 2: Elimination Using Subtraction Solve by elimination. # 2x + y = 5 $ % 2x 5y =13
12 B. Example 2: Elimination Using Subtraction Solve by elimination. # 3x + 3y =15 $ % 2x + 3y = 5
13 C. Example 2: Elimination Using Subtraction Solve by elimination. # 3x + 4y =18 $ % 2x + 4y = 8
14 Example 2: Elimination Using Subtraction Solve by elimination. " x 10y = 60 # $ x +14y =12 a. b.! 5x + y = 0 " # 5x + 2y = 30 c. " 5x + 7y =11 # $ 5x + 3y =19
15 When two equations each contain the same term, you can subtract one equation from the other to solve the system. To subtract an equation add the opposite of each term.
16 In some cases, you will first need to multiply one or both of the equations by a number so that one variable has opposite coefficients. This will be the new Step 1.
17 A. Example 3: Elimination Using Multiplication First Solve the system by elimination. # x + 2y =11 $ % 3x + y = 5
18 B. Example 3: Elimination Using Multiplication First Solve the system by elimination. # 5x + 2y = 32 $ % 2x + 3y =10
19 C. Example 3: Elimination Using Multiplication First Solve the system by elimination. # 3x + 2y = 6 $ % x + y = 2
20 Example 3: Elimination Using Multiplication First Solve the system by elimination. " 2x + 3y =12 # $ 5x y =13 a. b. " 3x + 4y =12 # $ 2x + y = 8 c. " 2x + 4y = 4 # $ 3x + 5y = 3
21 Example 4: Application A) Paige has $7.75 to buy 12 sheets of felt and card stock for her scrapbook. The felt costs $0.50 per sheet, and the card stock costs $0.75 per sheet. How many sheets of each can Paige buy?
22 Example 4: Application B) Sam spent $24.75 to buy 12 flowers for his mother. The bouquet contained roses and daises. How many of each type of flower did Sam buy? Roses Daisies $2.50 $1.75
23 Example 4: Application a)sally spent $14.85 to buy 13 flowers. She bought lilies, which cost $1.25 each, and tulips, which cost $0.90 each. How many of each flower did Sally buy? b)each family in a neighborhood is contributing $20 worth of food to the neighborhood picnic. The Harlin family is bringing 12 packages of buns. The hamburger buns cost $2.00 per package. The hotdog buns cost $1.50 per package. How many packages of each type of bun did they buy?
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25 Lesson Quiz Solve each system by elimination. 1. 2x + y = 25 3y = 2x x + 4y = 18 x = 2y 4 2x + 3y = 15 3x + 2y = Harlan has $44 to buy 7 pairs of socks. Athletic socks cost $5 per pair. Dress socks cost $8 per pair. How many pairs of each can Harlan buy?
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