Objective:Objective: Solve a system of two linear Solve a system of two linear equations in two variables by equations in two variables by

elimination.elimination.

Standard:Standard: 2.8.11.H. Select and use an 2.8.11.H. Select and use an appropriate strategy to solve appropriate strategy to solve

systems of equations.systems of equations.

3.2 Solving Systems by Elimination

I. Elimination MethodThe 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 = 8

12 + y = 8

y = - 4

Solution is (6, - 4)

CI

b. 2x + 5y = 15

–4x + 7y = -13

c. 4x – 3y = 15 8x + 2y = -10

8x + 2y = -10

-8x + 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 andsold if the production budget is $930 and the

expected revenue is $1920?

SmallSmall LargeLarge TotalTotal

ProductionProduction

CostCost

$5.50$5.50 $7.50$7.50 930930

Selling Selling PricePrice

$12$12 $15$15 19201920

5.5x + 7.5y = 93012x + 15y = 1920 * Multiply by -2 -11x – 15y = -1860 12x + 15y = 1920 x= 60 small y = 80 large

II. Dependent and Inconsistent SystemsEx 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 = 18

-10x + 6y = - 16 Multiplied by - 2

10x – 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 = -30

-2y + 5x = 15

4y – 10x = -30

- 4y + 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 = - 60

26x = - 52

x = -2

y = 4 (-2, 4) Consistent independent

Writing ActivitiesWriting Activities

3.2 Lesson Quiz