Solving Linear Systems – Linear Combination
Elimination Method
Name: _______________________________
Part I: Getting Started
A cafeteria sells fresh fruit by the weight. All apples weigh
the same and all oranges weigh the same. Use the picture
on the right to help you determine the weight of an orange
and the weight of an apple. Show your work or thinking
below.
Write a system of two equations to model each scale. Define your variables.
Represent your equations using a picture:
Represent your pictures algebraically:
I would use Linear Combination (the Elimination Method) when….
PRACTICE: Solve each system using the Elimination Method. Show your work!
Think: How can I add/or subtract these equations to eliminate one variable?
Part II: Linear Combination / Elimination: The Process!
(1) Align the System
See if the variables are linedup and ready to combine
“stacked”
(2) Make Opposite Coefficients
Multiply everything in
either/both equation(s) so that
one of the variables has
matching or opposite
coefficients
(3) Combine the System
Add or subtract the equations
together to “eliminate a
variable”
(4) Solve for Variable 1
Now that there is only one
variable in your equation,
solve for it
(5) Substitute and Solve for
Variable 2
Use the value you found in (4)
to find the other!
(6) Check your Solution
Evaluate both equations with
your solution
Example #1
{
−𝑥 + 𝑦 = 4
3𝑦 + 𝑥 = 4
Example #2
20𝑥 − 4𝑦 = 0
{
−3𝑥 − 𝑦 = −24
Part III: Linear Combination – Creating Opposite Coefficients (1-Step)
Use the six-step system to solve the following systems of equations:
(1)
(2)
(3)
(4)
(5)
(6)
Align the system
Make Opposite Coefficients
Combine the System
Solve for Variable I
Substitute and Solve for Variable 2
Check your Solution
**Write your solution as an ordered pair!!!!
1. {
3. {
−3𝑥 − 8𝑦 = 20
−10𝑥 + 2𝑦 = 38
3𝑥 + 2𝑦 = 1
4𝑥 + 3𝑦 = −2
2. {
−5𝑥 + 𝑦 = −2
6𝑦 − 3𝑥 = −12
4. {
6𝑥 − 4𝑦 = 0
8𝑥 + 3𝑦 = 25
Part IV: Real-Life Application
#1) 3 slices of pepperoni pizza plus 4 slices of a ham-and-pineapple pizza contains 930 Calories. 2 slices of
pepperoni pizza plus 2 slices of ham-and-pineapple pizza contains 540 Calories. How many Calories are in a
slice of pepperoni pizza? How many Calories are in a slice of ham-and-pineapple pizza?
a. Write a system of equations to represent this situation. Remember to define your variables and
label your problem conditions.
b. In words, how you can eliminate one of the variables to solve the system of equations?
c. Solve the system of equations using the linear combination /elimination method. Show your
work! Be sure to write your solution as an ordered pair.
d. Explain the solution in context: how many calories are in each kind of pizza?
Part IV: Real-Life Application
#2) Odell and his friends went to White Castle and spent $26 total on two Crave-Cases of burgers
and four orders of fries. Meanwhile, Brian and his friends also went to White Castle and spent $36
total on three Crave-Cases of burgers and five orders of fries. What is the cost of a crave case?
What is the cost of an order of fries?
a. Write a system of equations to represent this situation. Be sure to define your variables and label your problem
conditions.
b. Solve the system of equations using the linear combination / elimination method. Show your
work and write your solution as an ordered pair.
c. What does the solution mean in the context of the problem?
PRACTICE: Solve using Linear Combination /the Elimination Method. Show all steps of your
work. List your solution as an ordered pair.
−3𝑥 − 5𝑦 = −7
1. {
−4𝑥 + 5𝑦 = 14
2. {
3𝑥 + 4𝑦 = 24
6𝑥 + 8𝑦 = 24
3. {
−10𝑥 + 2𝑦 = −6
6𝑥 − 16𝑦 = 48
4. {
−5𝑥 − 8𝑦 = 17
5. {
2𝑥 − 7𝑦 = −17
6. {
−2𝑥 − 𝑦 = −20
6𝑥 − 5𝑦 = 12
−2𝑥 + 6𝑦 = 6
−4𝑥 + 12𝑦 = 12
Part V: Evaluating Methods of Solving
Fill in the table below to evaluate ALL solution methods we have discussed! 
Method of
Solving
Guess and
Check
Graphing
Table
Substitution
Linear
Combination
(Elimination)
Advantages
(Use this when…)
Disadvantages
(Do NOT use this when…)