7.2, 7.3 Solving by Substitution and Elimination

Solving by Substitution 1) Use one equation to get either x or y by itself Substitute what we just found in step 1 to the other equation Solve for one unknown Substitute the answer from step 3 to either of the original equation to solve for the other unknown Check your answer

x + y = 4 x = y + 2 Step 1) x is already by itself x = y + 2 Step 2) Substitute to the first equation x + y = 4 (y + 2) + y = 4 Step 3) 2y + 2 = 4 2y = 4 – 2 = 2 y = 1 Step 4) x = y + 2 x = 1 + 2 = 3 Solution: (3, 1)

2x + y = 6 3x + 4y = 4 Step 1) get y by itself 2x + y = 6 so y = -2x + 6 Step 2) Substitute to the second equation 3x + 4y = 4 3x + 4(-2x + 6) = 4 Step 3) 3x – 8x + 24 = 4 -5x + 24 = 4 -5x = -20 x = 4 Step 4) 2x + y = 6 2(4)+y = 6 so 8 + y = 6, y = -2 Solution: (4, -2)

More practice: (Make sure you check your answers) 1) 3x – 4y = 14 5x + y = 8 Answer: (2,-2) 2) 2x – 3y = 0 -4x + 3y = -1 Get x by itself: 2x – 3y = 0 so x = (3/2) y Substitute into: -4x + 3y = -1 -4(3/2)y + 3y = -1 -6y + 3y = -1 - 3y = -1 so y = 1/3 Solve for x: Use the first equation 2x – 3y = 0 2x – 3(1/3) = 0 2x – 1 = 0 x = ½ Answer: (1/2 , 1/3)

Example 1 The perimeter of a basketball court is 288 ft. The length is 40 ft longer than the width. Find the dimensions of the court.

Example 2 The sum of two numbers is 51. One number is 27 more than the other. Find the numbers.

Solve by Elimination Multiply some numbers to either or both equations to get 2 opposite terms (For example: 2x and -2x) Add equations to eliminate one variable Solve for 1 unknown Substitute the answer from step 3 to either of the original equation to solve for the other unknown Check your answer

Solve by elimination 2x – 3y = 0 -4x + 3y = -1 Ignore step 1 since we already have 2 opposite terms -3y and 3y Step 2 &3: Add 2 equations to eliminate y and solve for x -2x = -1 x = 1/2 Step 4: Choose the first equation 2x – 3y = 0 2(1/2) – 3y = 0 1 - 3y = 0 - 3y = -1 y = 1/3 Answer: ( 1/2, 1/3)

Ex2) 2x + 2y = 2 3x – y = 1 Step 1: Multiply 2 to the second equation 2x + 2y = 2 (2) 3x – y = 1 6x – 2y = 2 8x = 4 (step 2 and 3) x = 1/2 Step 4: Choose 3x – y = 1 3(1/2) – y = 1 3/2 - y = 1 - y = 1 – (3/2) = - ½ y = ½ Answer: (1/2, 1/2)

Ex3) 2x + 3y = 8 -3x + 2y = 1 Step 1: Multiply 3 to the 1st equation and 2 to the 2nd equation (3) 2x + 3y = 8 (2) -3x + 2y = 1 6x + 9y = 24 -6x + 4y = 2 13y = 26 (step 2 and 3) y = 2 Step 4: Choose -3x + 2y = 1 -3x + 2(2) = 1 -3x + 4 = 1 -3x = 1 -4 = -3 x = 1 Answer: (1,2)

Practice Solve by elimination 1) 3x – 4y = 14 5x + y = 8 Answer: (2,-2) 3x + 2y = 7 6x + 4y = 14 Answer: Infinite many solutions