Module 2 Review
Name:_______________ Date:_______ Hour:____
“Module 2 Review worksheet”
Create linear inequalities in two variables
1. Write the linear inequalities to represent each situation. Remember to define your variables.
a) A company has the space to build at most 36 computers every day. A desktop computer takes 1.5
hours to assemble, and a laptop computer takes 2 hours to assemble. They have 7 employees that can work a
maximum of 8 hours per day. They want to make at least 24 computers every day to keep up with demand
D: number of desktop computers; L: number of laptop computers
space: D + L ≤ 36
hours: 1.5D + 2L ≤ 7(8)
quantity: D + L ≥ 24
b) Graph each inequality and shade the feasible region.
c) List at least two solutions that use most of the time and space available.
laptop
feasible solutions
desktop
20
Graph linear inequalities in two variables
shade this side
2. Show all possible solutions to the equation 5𝑥 − 3𝑦 ≤ 15 on the graph below.
Label the x-intercept and y- intercept. Label on graph: (-5, 0) & (0, 3)
5
-5
5
-5
3. Show all possible solutions to the inequality −4𝑥 + 𝑦 < 5 on the graph below. Line should be dashed.
5
-5
5
-5
Solve systems of linear equations by graphing
4. Show all possible solutions to the system of inequalities {
𝑥 − 3𝑦 > 4
on the graph below.
−2𝑥 + 3𝑦 ≤ −6
5
Line should be dashed.
-5
5
-5
Determine whether an ordered pair is a solution to an equation, inequality, or system
5. The point (0 , 0) is a solution to which of the following system(s)?
(Select all that apply) b only
𝑥 + 6𝑦 = 4
𝑥 + 6𝑦 ≤ 4
a. {
b. {
−2𝑥 − 3𝑦 = 5
−2𝑥 − 3𝑦 < 5
d. {
𝑥 + 6𝑦 ≥ 4
−2𝑥 − 3𝑦 ≤ 5
𝑥 + 6𝑦 < 4
c. {
−2𝑥 − 3𝑦 ≥ 5
𝑥 + 6𝑦 > 4
e. {
−2𝑥 − 3𝑦 > 5
14 13
, ) is a solution to which of the following system(s)?
3 9
(Select all that apply) a and d
𝑥 + 6𝑦 = 4
𝑥 + 6𝑦 ≤ 4
𝑥 + 6𝑦 < 4
a. {
b. {
c. {
−2𝑥 − 3𝑦 = 5
−2𝑥 − 3𝑦 < 5
−2𝑥 − 3𝑦 ≥ 5
6. The point ( 
d. {
𝑥 + 6𝑦 ≥ 4
−2𝑥 − 3𝑦 ≤ 5
𝑥 + 6𝑦 > 4
e. {
−2𝑥 − 3𝑦 > 5
Rearrange equations to highlight a quantity of interest and interpret the parts of an equation in terms of the
context.
7. Convert the following equations: Standard Form to/from Slope-Intercept Form.
Standard Form
Slope-intercept Form
a.
6𝑥 − 2𝑦 = 11
y = 3x – 5.5
b.
2x + 3y = 12
𝑦 = −3𝑥 + 4
2
8. Chris is buying candy. His favorites are Twix and Skittles. The Twix are $0.75 and the Skittles are $0.60 and
he has $12 to spend on candy.
a. Write an inequality to represent this situation. 𝟎. 𝟕𝟓𝑻 + 𝟎. 𝟔𝟎𝑺 ≤ 𝟏𝟐
b. What does the slope of the border mean in terms of the context?
𝟓
If you solved for S, 𝑺 ≤ − 𝟒 𝑻 + 𝟐𝟎, the slope means that if he buys 5 fewer Skittles he can buy 4 more Twix.
𝟒
If you solved for T, 𝑻 ≤ − 𝟓 𝑺 + 𝟏𝟔, the slope means that if he buys 4 fewer Twix he can buy 5 more Skittles.
c. What does the x-intercept mean in terms of the context?
If you solved for S – the x-intercept is the number of Twix he can buy if he doesn’t buy any Skittles.
If you solved for T – the x-intercept is the number of Skittles he can buy if he doesn’t buy any Twix.
Solve systems of linear equations by symbol manipulation
9. Solve the following system of equations using 2 methods of your choosing. A graph has been provided in
case you have chosen that method, though you do not have to use it.
Solution: (-3, 2)
{
2𝑥 + 5𝑦 = 4
−𝑥 − 2𝑦 = −1
Substitution
Graphing
5
-5
5
Elimination
-5
10. Solve the following system of equations using 2 methods of your choosing. A graph has been provided in
case you have chosen that method, though you do not have to use it.
Solution: (4, -1)
{
Graphing
2𝑥 + 5𝑦 = 3
𝑥 + 6𝑦 = −2
Substitution
5
-5
5
Elimination
-5
Matrices (Systems 13H – Systems 14H)
11. Solve the following system of equations using 3 different methods. One of them MUST be using Matrix
Row Reduction.
Solution: (4, 3)
−𝑥 + 𝑦 = −1
{
3𝑥 − 5𝑦 = −3
Graphing
Substitution
Elimination
Matrix Row Reduction
(Label each step you take)
[
|
]
12. A toy store worker packed two boxes of identical dolls and plush toys for shipping in boxes that weigh 1
ounce when empty. One box held 3 dolls and 4 plush toys. The worker marked the weight as 12 oz. The
other box held 2 dolls and 3 plush toys. The worker marked the weight as 10 oz. Explain why the worker must
have made a mistake.
3D + 4P = 11
2D + 3P = 9
The solution to this system is not possible: plush toys weigh 5 oz. and dolls have a negative weight.
13. What do the substitution method and the elimination method have in common? Explain. Give an example
of a system that you would prefer to solve using one method instead of the other. Justify your choice.
answers vary
In both methods you are combining two equations to get rid of a variable so that you have a new equation
with one variable that can be solved.
14. Are the row operations for matrices more like the substitution method or the elimination method?
Explain.
answers vary, I think row operations for matrices are very similar to the elimination method because you
are trying to eliminate a number from each row by making an entry 0.
0 1 5
15. A student says that the matrix [
] shows that the solution of the system is (5, 3). What is the
1 0 3
student’s error? What is the correct solution of the system?
The correct solution is (3, 5). The student didn’t notice that the 0’s and 1’s were not in the standard
locations. The first row 0, 1, 5 means if there are 0 x’s and 1 y the solution is 5.