Name:
Date Due:
Secondary Math I – 6.2 Assignment: Using Linear Combinations to Solve a Linear System
Period:
Mr. Heiner
1. The two high schools in the Jefferson Hills School District are named Jefferson Hills East and Jefferson Hills West.
Both schools are taking field trips to the state capital. A total of 408 students from Jefferson Hills East will be
going in 3 vans and 6 buses. A total of 516 students from Jefferson Hills West will be going in 6 vans and 7 buses.
Each van has the same number of passengers and each bus has the same number of passengers.
a. Write an equation in standard form that represents the students from Jefferson Hills East. Let x
represent the number of students in each van, and let y represent the number of students in each bus.
b. Write an equation in standard form that represents the students from Jefferson Hills West. Use the
same variables as those used in part (a).
c. How are the equations in parts (a) and (b) the same? How are they different?
d. Describe the first step needed to solve the system using the linear combinations method. Identify the
variable that will be eliminated as well as the variable that will be solved for when you add the
equations.
e. Use your answer from part (d) to solve for one of the variables in the linear system of equations. Then
solve for the other variable. Show your work.
f.
What is the solution of the linear system? Interpret the solution of the linear system in terms of the
problem situation.
g. Check your solution algebraically.
Write a system of equations to represent each problem situation. Solve the system of equations using the linear
combinations (elimination) method.
Ex. The high school marching band is selling fruit baskets as a fundraiser. They sell a large basket containing 10 apples
and 15 oranges for $20. They sell a small basket containing 5 apples and 6 oranges for $8.50. How much is the marching
band charging for each apple and each orange?
Let 𝑥 represent the amount charged for each apple and 𝑦 represent the amount charged for each orange.
10𝑥 + 15𝑦 = 20
{
5𝑥 + 6𝑦 = 8.50
10𝑥 + 15𝑦 = 20
−2(5𝑥 + 6𝑦 = 8.50)
10𝑥 + 15𝑦 = 20
−10𝑥 − 12𝑦 = −17
3𝑦 = 3
𝑦=1
10𝑥 + 15(1) = 20
10𝑥 + 15 = 20
10𝑥 = 5
𝑥 = 0.5
The solution is (0.5, 1). The band charges $0.50 for each apple and $1.00 for each orange.
2. Asna works on a shipping dock at a tire manufacturing plant. She loads a pallet with 4 Mudslinger tires and 6
Roadripper tires. The tires on the pallet weigh 212 pounds. She loads a second pallet with 7 Mudslinger tires and
2 Roadripper tires. The tires on the second pallet weigh 184 pounds. How much does each Mudslinger tire and
each Roadripper tire weigh?
3. The Pizza Barn sells one customer 3 large pepperoni pizzas and 2 orders of breadsticks for $30. They sell another
customer 4 large pepperoni pizzas and 3 orders of breadsticks for $41. How much does the Pizza Barn charge for
each pepperoni pizza and each order of breadsticks?
Solve each system of equations using the linear combinations method.
Ex. {
3𝑥 + 5𝑦 = 8
the solution is (6, −2).
2𝑥 − 5𝑦 = 22
3𝑥 + 5𝑦 = 8
2𝑥 − 5𝑦 = 22
5𝑥 = 30
𝑥=6
10𝑥 − 6𝑦 = −6
5. {
5𝑥 − 5𝑦 = 5
4𝑥 − 𝑦 = 2
4. {
2𝑥 + 2𝑦 = 26
3(6) + 5𝑦 = 8
18 + 5𝑦 = 8
5𝑦 = −10
𝑦 = −2
2𝑥 − 4𝑦 = 4
6. {
−3𝑥 + 10𝑦 = 14
1.5𝑥 + 1.2𝑦 = 0.6
7. {
0.8𝑥 − 0.2𝑦 = 2
8.
𝑥 − 𝑦 = 11
9. {
2𝑥 + 𝑦 = 19
8𝑥 + 𝑦 = −16
10. {
−3𝑥 + 𝑦 = −5
3
1
3
𝑥 + 2𝑦 = −4
4
{ 2
2
2
𝑥 + 3𝑦 = 3
3
−0.2𝑥 − 0.9𝑦 = −2.5
11. {
−4𝑥 − 9𝑦 = −23
12. {
−6𝑥 + 6𝑦 = 6
13. {
−6𝑥 + 3𝑦 = −12
7𝑥 + 2𝑦 = 24
14. {
8𝑥 + 2𝑦 = 30
−0.4𝑥 − 1.5𝑦 = −1.7
15. {
−0.01𝑥 + 0.05𝑦 = −0.13
−𝑥 − 7𝑦 = 14
16. {
−4𝑥 − 14𝑦 = 28
−7𝑥 − 8𝑦 = 9
17. {
−4𝑥 + 9𝑦 = −22
3𝑥 − 2𝑦 = 2
18. {
5𝑥 − 5𝑦 = 10
3
1
1
− 5 𝑥 + 2 𝑦 = 10
6𝑥 + 4𝑦 = −10