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3.2 Solving Systems by Elimination
Objective:
Solve a system of two linear
equations in two variables by
elimination.
Standard:
2.8.11.H. Select and use an
appropriate strategy to solve
systems of equations.
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
Solution is (6, - 4)
3x = 18
x=6
2(6) + y = 8
12 + y = 8
y=-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 and
sold if the production budget is $930 and the
expected revenue is $1920?
Production
Cost
Small
$5.50
Large
$7.50
Total
930
$12
$15
1920
Selling
Price
5.5x + 7.5y = 930
12x + 15y = 1920
* Multiply by -2
-11x – 15y = -1860
12x + 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 = 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
Multiplied by – 1
-6x + 2y = -9
6x – 2y = 7
0=2
Empty Set
Inconsistent
Parallel both equations have a
slope m = 3
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 Activities
3.2 Lesson Quiz