Objective: Solve a system of two linear equations in two variables by elimination. Standard: H. Select and use an appropriate strategy to solve systems of equations. 3.2 Solving Systems by Elimination

I. Elimination Method The elimination method involves multiplying and combining the equations in a system in order to eliminate a variable. 1. Arrange each equation in standard form, Ax + By = C. 2. If the coefficients of x (or y) are the same number, use subtraction. 3. If the coefficients of x (or y) are opposites, use addition. 4. If the coefficients are different, multiply one or both to make them the same or opposite numbers. Then use step 2 or 3 to eliminate the variable. 5. Use substitution to solve for the remaining variable.

I.Independent Systems Ex 1. Use elimination to solve the system. Check your solution. a. 2x + y = 8 x – y = 10 3x = 18 x = 6 2(6) + y = y = 8 y = - 4 Solution is (6, - 4) CI

b. 2x + 5y = 15 –4x + 7y = -13

c. 4x – 3y = 15 8x + 2y = x + 6y = -30 Multiplied by - 2 8y = - 40 Y = -5 X = 0 Solution (0, -5) CI

Ex 2. This table gives production costs and selling prices per frame for two sizes of picture frames. How many of each size should be made and sold if the production budget is $930 and the expected revenue is $1920?SmallLargeTotal ProductionCost$5.50$ Selling Price $12$ x + 7.5y = x + 15y = 1920 * Multiply by x – 15y = x + 15y = 1920 x= 60 small y = 80 large

II. Dependent and Inconsistent Systems Ex 1. Use elimination to solve the system. Check your solution. a. 2x + 5y = 12 2x + 5y = 15 ** Multiply by – 1 to first equation -2x – 5y = -12 2x + 5y = 15 0 = 3 Empty Set Inconsistent Parallel Lines (both equations have a slope of -2/5)

b. -8x + 4y = -2 4x – 2y = 1 -8x + 4y = - 2 8x - 4y = 2 Multiplied by 2 0 = 0 ∞ Consistent Dependent

c. 5x - 3y = 8 10x – 6y = x + 6y = - 16 Multiplied by x – 6y = 18 0 = 2 Empty Set Inconsistent Parallel Lines (both equations have slope m = 5/3)

III. Independent, Dependent and Inconsistent Systems a. 6x – 2y = 9 6x – 2y = 7 0 = 2 Empty Set Inconsistent Parallel both equations have a slope m = 3 Multiplied by – 1 -6x + 2y = -9 6x – 2y = 7

b. 4y + 30 = 10x 5x – 2y = 15 4y – 10x = y + 5x = 15 4y – 10x = y + 10x = 30 Multiplied by 2 0 = 0 ∞ Consistent dependent

c. 5x + 3y = 2 2x + 20 = 4y 4(5x + 3y) = 4(2) 3(2x – 4y ) = 3(-20) 20x + 12y = 8 6x – 12y = x = - 52 x = -2 y = 4 (-2, 4) Consistent independent

Writing Activities

3.2 Lesson Quiz