Solving Systems of Equations by Elimination (6-3, 6-4) Objective: Solve systems of equations by using elimination with addition, subtraction, and multiplication. Solve real-world problems involving systems of equations.

Elimination Using Addition Using addition or subtraction to solve a system is called elimination. Use the following steps to solve a system by elimination: 1. Write the system so like terms with the same or opposite coefficients are aligned. 2. Add or subtract the equations, eliminating one variable. Then solve the equation. 3. Substitute the value from Step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair (x, y).

Example 1 Use elimination to solve the system of equations. -3x + 4y = 12 3x – 6y = 18 -3x + 4y = 12 +3x – 6y = 18 -2y = 30 -3x + 4(-15) = 12 -3x – 60 = 12 -3x = 72 y = -15 (-24, -15) +60 x = -24

Check Your Progress Choose the best answer for the following. Use elimination to solve the system of equations. 3x – 5y = 1 2x + 5y = 9 A.(1, 2) B.(2, 1) C.(0, 0) D.(2, 2) 3x – 5y = 1 2x + 5y = 9 5x = 10 x = 2 3(2) – 5y = 1 6 – 5y = y = -5 y = 1

Solving Systems - Elimination We can use elimination to find specific numbers that are described as being related to each other.

Example 2 Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers. 4x – 3y = 12 2x + 3y = 6 6x = 18 x = 3 4(3) – 3y = – 3y = y = 0 y = 0 The numbers are 3 and 0.

Check Your Progress Choose the best answer for the following. Four times one number added to another number is -10. Three times the first number minus the second number is -11. Find the numbers. A.-3, 2 B.-5, -5 C.-5, -6 D.1, 1 4x + y = -10 3x – y = -11 7x = -21 x = -3 4(-3) + y = y = y = 2

Elimination Using Subtraction Sometimes we can eliminate a variable by subtracting one equation from another.

Example 3 Use elimination to solve the system of equations. 4x + 2y = 28 4x – 3y = 18 5y = 10 y = 2 4x + 2(2) = 28 4x + 4 = x = 24 x = 6 (6, 2)

Check Your Progress Choose the best answer for the following. Use elimination to solve the system of equations. 9x – 2y = 30 x – 2y = 14 A.(2, 2) B.(-6, -6) C.(-6, 2) D.(2, -6) 9x – 2y = 30 x – 2y = 14 8x = 16 x = 2 9(2) – 2y = – 2y = y = 12 y = -6

Example 4 A hardware store earned $ from renting ladders and power tools last week. The store charged customers for a total of 36 days for ladders and 85 days for power tools. This week the store charged 36 days for ladders, and 70 days for power tools, and earned $829. How much does the store charge per day for ladders and for power tools? 36x + 85y = x + 70y = y = y = x + 85(8.50) = x = x = 236 x = 6.50 $6.50 per day for ladders and $8.50 per day for power tools.

Check Your Progress Choose the best answer for the following. For a school fundraiser, Marcus and Anisa participated in a walk-a-thon. In the morning, Marcus walked 11 miles and Anisa walked 13. Together they raised $ After lunch, Marcus walked 14 miles and Anisa walked 13. In the afternoon they raised $ How much did each raise per mile of the walk-a-thon? A.Marcus: $22.00, Anisa: $21.65 B.Marcus: $21.00, Anisa: $22.50 C.Marcus: $24.00, Anisa: $20.00 D.Marcus: $20.75, Anisa: $ x + 13y = x + 13y = x = -63 x = 21 11(21) + 13y = y = y = y = 22.50

Elimination Using Multiplication Sometimes neither variable can be eliminated by adding or subtracting. You can use multiplication to solve. Use the following steps to solve a problem by elimination that requires multiplication: 1. Multiply at least one equation by a constant to get two equations that contain opposite terms. 2. Add the equations, eliminating one variable. Then solve the equation. 3. Substitute the value from Step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair (x, y).

Example 5 Use elimination to solve the system of equations. 2x + y = 23 3x + 2y = 37 -2( )  -4x – 2y = -46  3x + 2y = 37 -x = -9 x = 9 2(9) + y = y = y = 5 (9, 5)

Check Your Progress Choose the best answer for the following. Use elimination to solve the system of equations. x + 7y = 12 3x – 5y = 10 A.(1, 5) B.(5, 1) C.(5, 5) D.(1, 1) -3( )  -3x – 21y = -36  3x – 5y = y = -26 y = 1 x + 7(1) = 12 x + 7 = x = 5

Elimination Method Sometimes you have to multiply each equation by a different number in order to solve the system.

Example 6 Use elimination to solve the system of equation. 4x + 3y = 8 3x – 5y = -23 5( ) 3( )  20x + 15y = 40  9x – 15y = x = -29 x = -1 4(-1) + 3y = y = y = 12 y = 4 (-1, 4)

Check Your Progress Choose the best answer for the following. Use elimination to solve the system of equations. 3x + 2y = 10 2x + 5y = 3 A.(-4, 1) B.(-1, 4) C.(4, -1) D.(-4, -1) 2( ) -3( )  6x + 4y = 20  -6x – 15y = y = 11 y = -1 3x + 2(-1) = 10 3x – 2 = x = 12 x = 4

Solve Real-World Problems Sometimes it is necessary to use multiplication before elimination in real-world problem solving too.

Example 7 A fishing boat travels 10 miles downstream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate in miles per hour of the boat in still water. x: rate of boat in still water. y: rate of current 30 minutes = ½ hour 40 minutes = 2 / 3 hour ½ (x + y) = 10 2 / 3 (x – y) = 10 2( ) 3 / 2 ( ) x + y = 20 x – y = 15 2x = 35 x = 17.5 mph

Check Your Progress Choose the best answer for the following. A helicopter travels 360 miles with the wind in 3 hours. The return trip against the wind takes the helicopter 4 hours. Find the rate of the helicopter in still air. A.103 mph B.105 mph C.100 mph D.17.5 mph 3(x + y) = 360 4(x – y) = 360 x + y = 120 x – y = 90 2x = 210