1. Solving Linear Systems by Linear Combinations
Objective:
Students will solve a linear system by
combinations ( Elimination ).

2. Algebra Standards:
8.EE.8 Students solve a system of two linear equations in two
variables algebraically, and are able to interpret the answer
graphically. Students are able to use this to solve a system
of two linear inequalities in two variables, and to sketch the
solution sets.

3. { 3 3 -4 -4 Solve the linear system 2x + 4y = 10 Check answer!
#1 Add the Equations
Solve the linear system {
2x + 4y = 10
Check answer!
x – 4y = 20
Solution:
2x + 4y = 10
x – 4y = 20
x – 4y = 20 10 – 4y
= 20 1 3x 1 = 30
-10
-10 1
-4y = 10 x = 10 5 y = – 5
Solution is (10, ) – 2 2

4. { 3 3 -4 -4 Solve the linear system 4x + y = -4 Check answer!
#2 Add the Equations
Solve the linear system {
4x + y = -4
Check answer!
-4x + 2y = 16
Solution:
4x + y = -4
-4x + 2y = 16
-4x + 2y = 16
-4x +2
(4)
=16 1 3y = 12
-4x
+ 8
=16 -8 -8 y = 4
-4x 1 = 8
Solution is (-2, 4) x = -2

5. { -1[ ] Solve the linear system x + 5y = 2 Check answer! x – 4y = 20
#3 Multiply then add
Solve the linear system {
x + 5y = 2
Check answer!
x – 4y = 20
Solution:
x + 5y = 2
x + 5y = 2
-1[ ]
x – 4y = 20 -x
+ 4y
= -20 1 9y =
-18
x + 5y = 2 y = -2 x
+ 5
(-2)
= 2 x
– 10
= 2
+10
+10
Solution is (12, -2) x = 12

6. { 5[ ] 4[ ] 23 23 Solve the linear system 3x + 4y = 6 Check answer!
#4 Multiply then add
Solve the linear system {
3x + 4y = 6
Check answer!
2x – 5y = -19
Solution:
5[ ]
3x + 4y = 6
15x
+ 20y
= 30
4[ ]
2x – 5y = -19 8x
– 20y
= -76 1
23x =
-46
3x + 4y = 6 3
(-2) + 4y
= 6 x = -2 -6 + 4y
= 6 +6 +6
Solution is (-2, 3) 1 4y = 12 y = 3

7. { -1[ ] Solve the linear system 3x + 2y = 8 Check answer! 2y = 12 – 5x
#5 Move the variable first then multiply and then add
Solve the linear system {
3x + 2y = 8
Check answer!
2y = 12 – 5x
Solution:
3x + 2y = 8
3x + 2y = 8
2y = 12 – 5x
+5x x
-1[ ] 5x
+ 2y
= 12
-5x
– 2y
= -12 1
-2x = -4 2y
+ 5x
= 12
3x + 2y = 8 3
(2)
+ 2y
= 8 x = 2 6
+ 2y
= 8
Solution is (2, 1) -6 -6 2y
= 2 1 y = 1

8. -6[ ] -2 -2 In one day the National Civil Rights Museum in Memphis,
#6 Word Problem
In one day the National Civil Rights Museum in Memphis,
Tennessee, admitted 321 adults and children and collected
$ The price of admission is $6 for an adult and $4 for a
child. How many adults and how many children were admitted
to the museum that day?
Solution:
How many
x = adults
Money
y = children 6x + 4y
= 1590 x + y
= 321
-6[ ] x + y
= 321
-6x
– 6y
= -1926 6x + 4y
= 1590 6x
+ 4y
= 1590
– 2y 1
= -336 x +
168
= 321
-168
-168
153 adults y =
168 x =
153
168 children

9. x + y = 58 x + y = 58 x – y = 16 37 + y = 58 2x = 74 -37 -37 2 2 y =
#7 Word Problem
Find two numbers whose sum is 58 and whose difference is 16.
Solution:
x = 1st number
y = 2nd number
x + y = 58 x + y
= 58 x – y
= 16 37 + y = 58 1 2x = 74
-37
-37 y = 21 x = 37

10. #8 Word Problem
The sum of two numbers is 56. The sum of one third of the
first number and one fourth of the second number is 16. Find
the numbers.
x = 1st number
Solution:
y = 2nd number x + y
= 56
-4[ ]
x + y = 56
-4x
– 4y =
-224 1 1
12[ ] x + y
= 16 4x + 3y
= 192
4x + 3y = 192 3 4
– y 1 =
-32 x
+32 = 56
-32
-32 y = 32 x = 24

11. -3[ ] How many gallons of 30% alcohol solution and how many
#9 Word Problem (Mixture)
How many gallons of 30% alcohol solution  and how many
of 60% alcohol solution must be mixed  to produce
18 gallons of 50% solution?
Solution:
Gallons of Solution
x = 30% Alchol Sol.
Solution(%)
y = 60% Alcohol Sol. x + y
= 18
10[ ]
0.3 x +
0.6 y =
(0.5) 18
18 = 50% Alcohol Sol. 3x + 6y =
(5) 18
-3[ ] x + y
= 18
-3x
– 3y
= -54 3x + 6y = 90 3x + 6y = 90 1 3y = 36 x + 12
= 18
-12
-12 y = 12 x = 6

12. Assignment
Book Pg. 290
# 1, 2, 4, 7, 19