6-4: Elimination using Multiplication

1) Use elimination to solve the system: 5x + y = 9 3x – y = 7 (2, -1) (-2, 1) (4, -3) (5, -2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

2) Use elimination to solve the system: 2x + 4y = -8 2x + y = 1 (3, -2) (2, -3) (1, 3) (-1, 2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Warmup Scores 4 Daniel Helper 3 Andrew Duncan Anya Augin Naomi Bervin Alexander Vendress

6-4: Elimination using Multiplication The difference of involving multiplication is highlighted below in red Write the system so like terms with the same or opposite coefficients are aligned Multiply at least one equation by a constant to get two equations that contain opposite terms Add or subtract the equations, eliminating one variable. Then solve the equation Substitute the value from Step 2 into one of the equations and solve for the other variable. Write the solutions in (x, y) order

6-4: Elimination using Multiplication Example of multiplying one equation 2x + y = 23 3x + 2y = 37 No opposites found, but multiply the top equation by -2 to eliminate the “y” -4x – 2y = -46 3x + 2y = 37 -x = -9 Divide by -1 x = 9 (continued next slide)

6-4: Elimination using Multiplication Example of multiplying one equation 2x + y = 23 3x + 2y = 37 x = 9 2(9) + y = 23 Substitute x = 9 18 + y = 23 Multiply 2(9) y = 5 Subtract 18 both sides (9, 5) is the solution

Ex 1: Use elimination to solve the system x + 7y = 12 3x – 5y = 10 (1, 5) (5, 1) (5, 5) (1, 1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

6-4: Elimination using Multiplication Multiplying both equations 4x + 3y = 8 3x – 5y = -23 No opposites found, but multiply the top equation by 5 and bottom equation by 3 to eliminate the “y” 20x + 15y = 40 9x – 15y = -69 29x = -29 Divide by 29 x = -1 (continued next slide)

6-4: Elimination using Multiplication Multiplying both equations 4x + 3y = 8 3x – 5y = -23 x = -1 4(-1) + 3y = 8 Substitute x = 9 -4 + 3y = 8 Multiply 4(-1) 3y = 12 Add 4 both sides y = 4 Divide both sides by 3 (-1, 4) is the solution

Ex 2: Use elimination to solve the system 3x + 2y = 10 2x + 5y = 3 (-4, 1) (-1, 4) (4, -1) (-4, -1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

6-4: Elimination using Multiplication Word Problem A personal airplane traveling with the wind flies 520 miles in 4 hours. On the return trip (against the wind), the airplane takes 5 hours to travel the same distance. Find the speed of the airplane without the wind. Let x be the speed of the airplane Let y be the speed of the wind Distance = rate ● time 520 = 4(x + y) 130 = x + y Multiply each side by 2 520 = 5(x – y) 104 = x – y Multiply each side by 3/2 (continued next slide)

6-4: Elimination using Multiplication Word Problem Let x be the speed of the boat Let y be the speed of the current 130 = x + y 104 = x – y Have opposites, simply add the equations 234 = 2x Divide by 2 117 = x The plane is traveling 117 miles/hour

A helicopter travels 360 miles with the wind in 3 hours A helicopter travels 360 miles with the wind in 3 hours. The return trip against the wind takes 4 hours. Find the rate of the helicopter in still air. 103 mph 105 mph 100 mph 17.5 mph 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Warm-up + In-Lesson Scores Participant 1 Participant 2 Participant 3 Participant 4 Participant 5

6-4: Elimination using Multiplication Assignment Page 359 – 360 1 – 23, odds