Warm Up #30: Solve by substitution

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Presentation transcript:

Warm Up #30: Solve by substitution (1) y = 2x + 7 2y – 5x = 11 x = 3y – 4 2x – y = 7 (2) x + 2y = 10 3x + 4y = 8 -3x + y = 5

8.3 The Addition Method Objective: To solve a system of equations by using the addition method

The Addition Method Write the second equation below the first. Add the equations together and solve for the remaining variable. Substitute that value into either of the original equations and solve the second variable. Used when the addition will remove one of the variables.

Solve: + x + y = 5 x - y = 1 x - y = 1 x - y = 1 x - y = 1 x - y = 1 = 6 x = 3 3 + y = 5 y = 2 (3,2)

x + y = 5 x + y = 5 2x - y = 4 3x = 9 x = 3 3 + y = 5 y = 2 (3,2)

2 Special Cases No solution Case 1: NO SOLUTION Both variables cancel out but the constant does not Leaving a false equation. 4x - 2y = 2 -4x + 2y = -16 0 + 0 = -14 No solution

Case 2: INFINITELY MANY SOLUTIONS Both variables and the constant cancel out Leaving a true equation 5x - 7y = 6 -5x + 7y = -6 0 + 0 = 0 Infinitely many solutions

They all do not cancel out so easily If the equations do not eliminate a variable when you add them together: Multiply the whole equation by a number that will help you cancel it out. (multiply every term in the equation by that number)

2 4x - 2y = 7 4x - 2y = 7 3x + y = 4 6x + 2y = 8 10x = 15 x = 1.5 (1.5,-0.5)

-1 2x + 3y = 8 2x + 3y = 8 x + 3y = 7 -x - 3y = -7 x = 1 2(1) + 3y = 8 (1,2)

3 4 4x + 2y = 18 12x + 6y = 54 -3x + 5y = 6 -12x + 20y = 24 26y = 78 (3,3) y = 3

3 -5 5x + 3y = 2 15x + 9y = 6 -15x - 35y = 20 3x + 7y = -4 -26y = 26 (1,-1) x = 1

The sum of two numbers is 72. The difference is 58. Find the numbers. x + y = 72 x - y = 58 7 and 65

The sum of the length and width of a rectangle is 25 cm The sum of the length and width of a rectangle is 25 cm. The length is 2 less than twice the width. Find the length and width.

Assignment: Page 371 (10-32) even