Algebra I
Linear Systems & Inequalities
Practice Test
Name__________________________
1. Writing: Use complete sentences and be specific.
a. From a table, how can you find a solution to a system of equations? When would it be beneficial/efficient
to use a table to solve a system of equations?
b. From a graph, how can you find a solution to a system of equations? When would it be useful/efficient to
use graphing as a method to solve a system?
c. When would it be useful/efficient to use linear combinations as a method to solve a system of
equations?
2. Solve each system of equations using the method of your choice. **You must use
each method at least once (graphing/table, elimination.) Show your work and
write your solution as an ordered pair. Check your solution.
𝑦 = −3𝑥 + 5
a. {
5𝑥 − 4𝑦 = −3
−4𝑥 − 2𝑦 = 12
b. {
𝑥 − 𝑦 = 15
c. {
𝑦 = 3𝑥 − 4
1
2
𝑦=− 𝑥+3
8𝑥 + 𝑦 = −16
d. {
3𝑥 = 5 + 𝑦
3. Solve the system of equations or inequalities by graphing. Identify your solution appropriately
as either an ordered pair (x,y) or a shaded region of the graph.
a. {
𝑦 = −2𝑥 + 1
𝑦 =𝑥−8
b. {
𝑦 ≥ −3𝑥 − 1
𝑦 < 8 − 3𝑥
Writing: How is graphing a system of EQUATIONS different than graphing a system of INEQUALITIES?
How does the solution look different?
4. Two cell phone plans offer differing text packages. Plan A had a $5.60 per month charge and charges
$0.03 per text. Plan B has no monthly charge, but a charge of $0.10 per text.
a) Is there a certain number of texts, when the two plans cost the same amount? Determine your answer
by setting up a system of equation that model the two plans. Be sure to define your variables!
b) How much will it cost to send this many text messages a month? Does it matter which plan that you
choose? Explain your answer.
5. In Lewis Carroll’s Through the Looking Glass, Tweedledum says, “The sum of your weight and twice
mine is 361 pounds.” Tweedledee replies, “The sum of your weight and twice mine is 362 pounds.” Find
both of their weights.
a. Write a system of equations to model this situation. Make sure to define your variables and label the
problem conditions!
b. Solve the system of equations and interpret your answer in the context of the situation.
6.
John mows yards for his dad’s landscaping business for $10 per hour, and he also works at a bakery for
$15 per hour. He can work at most 52 hours per week during the summer. He needs to make at least
$600 per week to cover his living expenses.
a. Let x represent the number of hours he works for the landscaping business and let y represent the
number of hours that he works at the bakery. Write a system of linear inequalities to represent
the situation (hint: Don’t forget to include “common sense inequalities”)
b. Construct a graph to represent the solution set.
c. If John works 14 hours mowing and 30 hours at the bakery, does this satisfy all of the problem’s
constraints?
d. If John must work a minimum of 10 hours for his father, will he be able to make enough money
to cover his living expenses? Show that work that leads to your answer.
e. Identify one possible combination of hours he works at the landscaping business and hours he
works at the bakery. Justify how you know from the graph.
f. Identify one combination of landscaping hours and bakery hours John CANNOT work. Justify
how you know from the graph.
7. Given the graph to the right, answer the following questions:
a. Write a system of linear inequalities to represent this graph.
{
b. Is the point (-4, 1) a solution to the system? Why or why not?
c. Is the point (-3, 5) a solution to the system? Why or why not?
8. Mrs. Gabel and Ms. Baucum are both solving the following system of equations:
𝒚 = 𝟓𝒙 − 𝟒
{
−𝟑𝒚 + 𝟏𝟓𝒙 = 𝟏𝟐
a. Mrs. Gabel says that there is NO solution and Ms. Baucum says that there are INFINITELY
MANY solutions. Which of your teachers is correct and why?
b. If the other teacher had been correct, what would it have looked like when she finished solving
the system?
c. Jeff is solving a system of equations using substitution. He did all of the work correctly, but
when he finished his algebra work, his final line said -3 = 8. What does this mean?
d. Emily is solving a system of equations using elimination. She did all of her algebraic work
correctly, but both variables dropped out and left the line 7 = 7. What does this mean?
9. What does it look like algebraically when there is ONE SOLUTION to a system? (Think about what
happens when you solve a system of equations and you get a solution).