Section 3: Solving Equations and
Inequalities with Two Variables
MAFS.912.A-REI.4.11 Explain why the 𝑥𝑥-coordinates of the
points where the graphs of the
equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥)
intersect are the solutions of the
equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions
approximately (e.g., using technology
to graph the functions, make tables of
values, or find successive
approximations). Include cases where
𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear,
polynomial, rational, absolute value,
exponential, and logarithmic functions.
This section only covers linear cases.
MAFS.912.A-REI.4.12 Graph the solutions to a linear
inequality in two variables as a halfplane (excluding the boundary in the
case of a strict inequality), and graph
the solution set to a system of linear
inequalities in two variables as the
intersection of the corresponding halfplanes.
MAFS.912.S-ID.3.7
Interpret the slope (rate of change)
and the intercept (constant term) of a
linear model in the context of the
data.
The following Mathematics Florida Standards will be covered
in this section:
MAFS.912.ACED.1.2
MAFS.912.ACED.1.3
MAFS.912.A-REI.3.5
MAFS.912.A-REI.3.6
Create equations in two or more
variables to represent relationships
between quantities; graph equations
on coordinate axes with labels and
scales.
Represent constraints by equations or
inequalities and by systems of
equations and/or inequalities, and
interpret solutions as viable or
nonviable options in a modeling
context. For example, represent
inequalities describing nutritional and
cost constraints on combinations of
different foods.
Prove that, given a system of two
equations in two variables, replacing
one equation by the sum of that
equation and a multiple of the other
produces a system with the same
solutions.
Solve systems of linear equations
exactly and approximately (e.g., with
graphs), focusing on pairs of linear
equations in two variables.
MAFS.912.A-REI.4.10 Understand that the graph of an
equation in two variables is the set of
all its solutions plotted in the
coordinate plane, often forming a
curve (which could be a line).
Videos in this Section
Video 1:
Video 2:
Video 3:
Video 4:
Video 5:
Video 6:
Video 7:
Video 8:
Solution Sets to Equations with Two Variables
Discovering Slope
Discovering Slope and 𝑌𝑌-intercept
Finding Solution Sets to Systems of Equations Using
Substitution and Graphing
Finding Solution Sets to Systems of Equations Using
Elimination
!
Why Does Elimination Work?
Solution Sets to Inequalities with Two Variables
Finding Solution Sets to Systems of Linear Inequalities
Section 3: Solving Equations and Inequalities with Two Variables
45
Section 3 – Video 1
Solution Sets to Equations with Two Variables
Consider 𝑥𝑥 + 2 = 5. What is the only possible value of 𝑥𝑥 that
makes the equation a true statement?
What do you notice about the points you graphed?
Are these the only solutions to the equation 𝑥𝑥 + 𝑦𝑦 = 5?
Now consider 𝑥𝑥 + 𝑦𝑦 = 5. What are some solutions for 𝑥𝑥 and
𝑦𝑦 that would make the equation true?
How many solutions are there to the equation 𝑥𝑥 + 𝑦𝑦 = 5?
Possible solutions can be listed as ordered pairs.
Let’s Practice!
Graph each of the ordered pairs from the previous problem
on the graph below.
Tammy has 10 songs on her phone’s playlist. The playlist
features songs from her two favorite artists, Beyoncé and
Pharrell. What are some possibilities for the number of songs
she could have from each artist?
Create an equation using two variables to represent this
situation. Explain the meaning of your equation.
List at least three solutions to the equation that you created.
Do we have infinitely many solutions to this equation? Why or
why not?
In this case, our solutions must be ________________________
numbers.
46
Section 3: Solving Equations and Inequalities with Two Variables
! Notice that the solutions follow a linear pattern. However,
they do not form a line.
! This is called a discrete function.
Try It!
The sum of two numbers is 15.
List at least two possible solutions.
Create a graph that represents the solution set to your
equation.
Create an equation using two variables to represent this
situation.
How many solutions are there to this equation?
Create a visual representation of all the possible solutions on
the graph.
Why are there only positive values on this graph?
How is this graph different from the graph in the previous
problem?
!
Do we have infinitely many solutions to this equation? Why or
why not?
Section 3: Solving Equations and Inequalities with Two Variables
47
In this case, our solutions must be ________________________
numbers.
! Because of this, the solutions will form a line.
! This is called a continuous function.
BEAT THE TEST!
1. Elizabeth’s tablet has a combined total of 20 apps and
movies. Let 𝑥𝑥 represent the number of apps and 𝑦𝑦 represent
the number of movies. Which of the following statements
represent the number of apps and movies on Elizabeth’s
tablet? Check all that apply.
"
"
"
"
"
"
What if the problem said the sum of two integers is 15?
Graph the solution. How is the graph different from the original
problem?
𝑥𝑥 + 𝑦𝑦 = 20
7 apps and 14 movies
𝑥𝑥 − 𝑦𝑦 = 20
𝑦𝑦 = −𝑥𝑥 + 20
8 apps and 12 movies
𝑥𝑥𝑥𝑥 = 20
Is this function discrete or continuous?
48
Section 3: Solving Equations and Inequalities with Two Variables
2. The graph shown below depicts the total cost of purchasing
apples at a certain price ($) per bushel.
Bushels of Apples
Section 3 – Video 2
Discovering Slope
Jada is reading The Fault in Our Stars and reads 16 pages
every day. Represent the situation on the graph below.
Cost in Dollars
Graph 1
Number of Bushels
Part A: The graph “Bushels of Apples” is
A
B
C
D
a continuous function.
a discrete function.
an inequality.
nonlinear.
Bianca started reading the same book. She read eight pages
every day. Represent the situation on the graph below using
the same scales on the axes that you used in the graph
above.
Graph 2
Part B: Each bushel of apples costs
A
B
C
D
$1.00.
$5.00.
$10.00.
This cannot be determined from the information
given.
!
Section 3: Solving Equations and Inequalities with Two Variables
49
Try It!
Aaron loves Cherry Coke. Each mini can contains 100 calories.
On the graph below, represent his caloric intake based on the
number of Cherry Coke mini cans he drinks.
Graph 3
In each of the graphs, we were finding the rate of change in
the given situation. What is the rate of change for each of the
graphs?
Graph 1: ____________________ per _______________________
Graph 2: ____________________ per _______________________
Graph 3: ____________________ per _______________________
Graph 4: ____________________ per _______________________
This is also called the ________________________ of the line.
Jacobe likes to munch on carrot snack packs. Each snack
pack contains 40 calories. On the graph below, represent
Jacobe’s caloric intake based on the number of carrot packs
he eats using the same scale in the graph above.
We can also find slope by looking at the
Miles
Hours
50
!!!"#$ !" !
or
!"#$
!"#
.
What is the slope of the following graph? What does the slope
represent?
Graph 4
!!!"#$ !" !
Section 3: Solving Equations and Inequalities with Two Variables
Try It!
What is the slope of the following graph? What does the slope
represent?
GPA
Souvenirs
Purchased Keisha’s Vacation
Souvenirs
Freedom High School collected data on the GPA of various
students and the number of hours they spend studying each
week. A scatterplot of the data is shown below with the line of
best fit. What is the slope of the line of best fit? What does it
represent?
Time Spent Studying Each Week
(Hours)
Days of Vacation
!
Section 3: Solving Equations and Inequalities with Two Variables
51
Sarah’s parents give her $100.00 allowance at the beginning
of each month. Sarah spends her allowance on comic books.
The graph below represents the amount of money Sara spent
on comic books last month. What is the slope? What does the
slope represent? How much does one comic book cost?
BEAT THE TEST!
1. A cleaning service cleans many apartments each day. The
following table shows the number of hours the cleaners
spend cleaning and the number of apartments they clean
during that time.
Amount of Allowance Left
(in Dollars)
Apartment Cleaning
Time (Hours)
Apartments
Cleaned
It doesn’t matter where the negative sign is
52
2 2
4 3
6 4
8 Part A: Represent the situation on the graph below.
Number of Comic
Books Purchased
1
placed; for example,
equivalent. !!
!
,
!
!!
, and −
!
!
are all
Section 3: Solving Equations and Inequalities with Two Variables
Part B: The data suggest a linear relationship between the
number of hours spent cleaning and the number of
apartments cleaned. Assuming the relationship is
linear, what does the rate of change represent in the
context of this relationship?
A The number of apartments cleaned after one
hour. B The number of hours it took to clean one
apartment. C The number of apartments cleaned each hour. D The number of apartments cleaned before the
company started cleaning. Section 3 – Video 3
Discovering Slope and 𝒀𝒀-Intercept
Cab fare includes an initial fee of $2.00 plus $3.00 for every
mile traveled.
Define the variables and write an equation that represents this
situation.
Represent the situation on a graph.
Part C: Which equation describes the relationship between
the time elapsed and the number of apartments
cleaned?
A
B
C
D
𝑦𝑦 = 𝑥𝑥 𝑦𝑦 = 𝑥𝑥 + 2
𝑦𝑦 = 2𝑥𝑥 𝑦𝑦 = 2𝑥𝑥 + 2 What is the slope of the line? What does the slope represent?
At what point does the line intersect the 𝑦𝑦-axis? What does this
point represent?
!
This point is commonly known as the 𝒚𝒚-intercept of a line.
Section 3: Solving Equations and Inequalities with Two Variables
53
Try It!
Consider the following graph:
You saved $250.00 to spend over the summer. You decide to
budget $25.00 to spend each week.
Total Cost
(in Dollars)
Define the variables and write an equation that represents this
situation.
Represent the situation on a graph.
Number of Visits to the
Community Pool
What is the slope of the line?
What is the 𝑦𝑦-intercept?
Define the variables and write an equation that represents this
situation.
What is the slope of the line? What does the slope represent?
What does the slope represent?
What is different about the slope of this problem compared to
our earlier problem? Why is it different?
What does each point represent?
What is the 𝑦𝑦-intercept? What does this point represent?
54
Section 3: Solving Equations and Inequalities with Two Variables
Consider the three equations that you wrote regarding the
cab ride, your summer spending habits, and the community
pool membership.
Graph 𝑦𝑦 = 2𝑥𝑥 + 3.
What do you notice about the constant term? What do you
notice about the coefficient of the variable?
! The constant term is the ________________________
________________________.
! The coefficient is the ________________________.
! This is called slope-intercept form of an equation.
We can use slope-intercept form to graph any linear equation.
Where does the graph intersect the 𝑦𝑦-axis?
What is the slope, or rate of change?
The coefficient of 𝑥𝑥 is the slope and the
constant term is the 𝑦𝑦-intercept ONLY if the
equation is in slope-intercept !form, 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏. Section 3: Solving Equations and Inequalities with Two Variables
55
Consider the equation 2𝑥𝑥 + 5𝑦𝑦 = 10.
How does this equation look different from slope-intercept
form of an equation?
Try It!
Graph the equation −4𝑥𝑥 − 5𝑦𝑦 = −10.
Rewrite the equation in slope-intercept form.
Identify the 𝑦𝑦-intercept.
Identify the slope.
Graph the equation.
56
Section 3: Solving Equations and Inequalities with Two Variables
BEAT THE TEST!
Amount of Money
Raised
(in dollars)
1. Line 𝑡𝑡, ∆𝐸𝐸𝐸𝐸𝐸𝐸, and ∆𝐹𝐹𝐹𝐹𝐹𝐹 are shown on the coordinate grid
below.
2. The senior class at Elizabeth High School was selling tickets to
raise money for prom. The graph below represents the
situation.
Number of Tickets Sold
Part A: How much does one ticket cost?
Which of the following statements are true? Check all that
apply.
" The slope of 𝐴𝐴𝐴𝐴 is equal to the slope of 𝐵𝐵𝐵𝐵.
" The slope of 𝐴𝐴𝐴𝐴 is equal to the slope of line 𝑡𝑡.
" The slope of line 𝑡𝑡 is equal to
" The slope of line 𝑡𝑡 is equal to
" The slope of line 𝑡𝑡 is equal to
!"
.
!"
!"
.
!"
!"
!"
Part B: How much money did the senior class have at the
start of the fundraiser?
.
" The 𝑦𝑦-intercept of line 𝑡𝑡 is 2.
" Line 𝑡𝑡 represents a discrete function.
Part C: What does the 𝑦𝑦-intercept represent?
!
Section 3: Solving Equations and Inequalities with Two Variables
57
Section 3 – Video 4
Finding Solution Sets to Systems of Equations
Using Substitution and Graphing
Brianna’s lacrosse coach suggested that she take yoga to
improve her flexibility. “Yoga-ta Try This!” Yoga Studio has two
membership plans. Plan A costs $20.00 per month plus $10.00
per class. Plan B costs $100.00 per month for unlimited classes.
Define the variables and write two equations to represent the
monthly cost of each plan.
Represent the two situations on the graph below.
What is the rate of change for each plan?
What does the rate of change represent in this situation?
What do the 𝑦𝑦-intercepts of the graphs represent?
Brianna is trying to determine which plan is more appropriate
for the number of classes she wants to attend.
When will the two plans cost exactly the same?
When is plan A the better deal?
When is plan B the better deal?
58
Section 3: Solving Equations and Inequalities with Two Variables
We can also help Brianna determine the best plan for her
without graphing. Consider our two equations again.
Represent the two situations on the graph below.
If 𝑦𝑦 represents the total cost, we simply want to know when the
total costs would be equal.
! Set the two plans equal to each other and solve for the
number of visits.
! This method is called solving by _________________________.
Vespa Scooter Rental rents scooters for $45.00 and $0.25 per
mile. Scottie’s Scooter Rental rents scooters for $35.00 and
$0.30 per mile.
Define the variables and write two equations to represent the
situation.
What is the rate of change of each line? What do they
represent?
What do the 𝑦𝑦-intercepts of each line represent?
When will renting a scooter from Vespa Scooter Rental cost
the same as renting a scooter from Scottie’s Scooter Rental?
!
Section 3: Solving Equations and Inequalities with Two Variables
59
Describe a situation when renting from Vespa Scooter Rental
would be a better deal than renting from Scottie’s Scooter
Rental.
Try It!
Chelsea and John were playing basketball. Chelsea had eight
points, and John had 19 points. Chelsea didn’t like losing to
John, so she changed the rules! For the rest of the game, she
would get two points for every basket she made, and John
would only get one point per basket. After the rules changed,
they scored the same number of baskets and ended the
game with a tie!
Define the variables and write two equations to represent
Chelsea’s and John’s potential scores after Chelsea changed
the rules.
Use the substitution method to help the renter determine when
the two scooter rentals will cost the same amount.
At the end of the game, Chelsea and John were tied. Use
graphing or substitution to determine how many baskets they
each scored after Chelsea changed the rules.
60
You should always check your solution by
plugging it back into both equations to make
sure that it gives you a true statement. Section 3: Solving Equations and Inequalities with Two Variables
BEAT THE TEST!
2. Rais and Erich are evaluating the system of equations shown
below:
1. In a basketball game, Jerry made 16 shots. Each of the shots
was worth either two or three points, and Jerry scored a total
of 39 points. Let 𝑥𝑥 represent the number of two-point shots
and 𝑦𝑦 represent the number of three-point shots. Write a
system of equations, in terms of 𝑥𝑥 and 𝑦𝑦, to model the
situation. Enter your equations in the space provided.
𝑦𝑦 = 𝑥𝑥 + 5 𝑦𝑦 = −3𝑥𝑥 + 1
Rais found the solution set (2, 7), while Erich found the
solution set (−1, 4).
Part A: How could Rais check to see if his set was correct
compared to Erich’s set?
Part B: Whose solution set is correct?
Part C: What did the other person do wrong? Explain.
!
Section 3: Solving Equations and Inequalities with Two Variables
61
Section 3 – Video 5
Finding Solution Sets to Systems of Equations
Using Elimination
Kevin and Pete are throwing a Super Bowl party. They are at
the store buying collectible items. Kevin is rooting for the
Seattle Seahawks, so he bought 20 cups and 30 plates with
the Seahawks logo. Pete is rooting for the New England
Patriots, and he bought 20 cups and 40 plates with the Patriots
logo. Kevin spent a total of $236.50, and Pete spent a total of
$289.00.
Let’s Practice!
Ruxin and Andre were invited to a Super Bowl party. They were
asked to bring pizzas and sodas. Ruxin bought three
pepperoni pizzas and four bottles of soda. Andre bought five
ham pizzas and two bottles of soda. Ruxin spent a total of
$48.05, and Andre spent a total of $67.25. Assuming the cost of
each pizza is the same and all the sodas cost the same
amount, write and solve a system of equations to determine
the cost of each pizza and bottle of soda.
Define the variables and write and solve a system of equations
to determine the amount spent on each plate and cup.
62
Always check to make sure that the variable
you solved for is actually answering the
question. Section 3: Solving Equations and Inequalities with Two Variables
Try It!
BEAT THE TEST!
Jazmin and Justine went shopping for back to school clothes.
Jazmin purchased three shirts and one pair of shorts. She spent
$38.00. Justine had more money to spend. She bought four
shirts and three pairs of shorts and spent $71.50.
Assuming all the shirts cost the same amount and all the shorts
cost the same amount, define the variables and write a
system of equations to represent each girl’s shopping spree.
1. Courtney works for the Red Cross and has been tasked with
buying non-perishable items for families recently displaced
by a hurricane. She finds a company willing to send her
Meals-Ready-to-Eat (MREs) at a discounted price. Cases of
small portion meals are $20.00 each and cases of large
portion meals are $30.00 each. She buys a total of 20 cases
and spends a total of $450.00. How many of each case did
she purchase?
Courtney purchased
and
cases of small portion meals
cases of large portion meals.
Use the elimination method to solve for the price of shorts.
!
Section 3: Solving Equations and Inequalities with Two Variables
63
Section 3 – Video 6
Why Does Elimination Work?
2. Fill in the boxes below to complete the following table.
Solve by Elimination:
Operations
2𝑥𝑥 − 3𝑦𝑦 = 8
3𝑥𝑥 + 4𝑦𝑦 = 46
Consider the following system of equations:
Equations
Labels
2𝑥𝑥 − 3𝑦𝑦 = 8
3𝑥𝑥 + 4𝑦𝑦 = 46
Equation 1
Equation 2
−6𝑥𝑥 + 9𝑦𝑦 = −24
New equation 1
Multiply equation 2 by 2.
𝑥𝑥 + 𝑦𝑦 = 4
𝑥𝑥 − 𝑦𝑦 = 6
The solution to this system is (5, −1). We can also see this when
we graph the lines.
New equation 2
−6𝑥𝑥 + 9𝑦𝑦 = −24
6𝑥𝑥 + 8𝑦𝑦 = 92 17𝑦𝑦 = 68
Divide by 17.
Solve for 𝑥𝑥.
Write 𝑥𝑥 and 𝑦𝑦 as
coordinates.
64
,
Solution to the
system
Section 3: Solving Equations and Inequalities with Two Variables
Let’s consider what happens when we multiply either of the
equations by some factor.
Let’s consider what happens when we add the two equations
in the system together.
What does the new equation look like on the graph of our
previous system?
Algebraically, show that (5, −1) is also a solution to the sum of
the two lines.
When you multiply an equation by some
factor, the resulting equation is equivalent to
the original. Therefore, it has the same solution
for the system. !
Section 3: Solving Equations and Inequalities with Two Variables
65
Let’s consider what happens when we subtract one of the
equations from the other.
Try It!
Consider the previous system:
What does the new equation look like on the graph of our
previous system?
Equation 1: 𝑥𝑥 + 𝑦𝑦 = 4
Equation 2: 𝑥𝑥 − 𝑦𝑦 = 6
Complete the following steps to show that replacing one
equation by the sum of that equation and a multiple of the
other equation produces a system with the same solutions.
Create a third equation by multiplying Equation 1 by two.
Create a fourth equation by finding the sum of the third
equation and Equation 2.
Show on the graph that the fourth equation has the same
solution as the original system.
Algebraically, show that (5, −1) is also a solution to the
difference of the two lines.
66
Section 3: Solving Equations and Inequalities with Two Variables
Let’s consider the following system, which has a solution of
(2, 5) and 𝑀𝑀, 𝑁𝑁, 𝑃𝑃, 𝑅𝑅, 𝑆𝑆, and 𝑇𝑇 are non-zero real numbers:
𝑀𝑀𝑀𝑀 + 𝑁𝑁𝑁𝑁 = 𝑃𝑃
𝑅𝑅𝑅𝑅 + 𝑆𝑆𝑆𝑆 = 𝑇𝑇
BEAT THE TEST!
1. Consider the following system:
Line P: 𝑥𝑥 + 3𝑦𝑦 = 12
Line Q: 2𝑥𝑥 + 𝑦𝑦 = −6
Create four equations that have the same solution as the
system.
Which of the following equations could be substituted for
line 𝑃𝑃 or line 𝑄𝑄 such that the new system would have the
same solution? Select all that apply.
"
"
"
"
"
2𝑥𝑥 + 6𝑦𝑦 = 12
3𝑥𝑥 + 4𝑦𝑦 = 6
3𝑥𝑥 + 9𝑦𝑦 = 36
−𝑥𝑥 + 2𝑦𝑦 = 18
4𝑥𝑥 + 2𝑦𝑦 = −6
!
Section 3: Solving Equations and Inequalities with Two Variables
67
2. The system
𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 = 𝐶𝐶
has the solution (1, −3), where
𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 = 𝐹𝐹
𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷, 𝐸𝐸, and 𝐹𝐹 are non-zero real numbers. Select all the
systems of equations with the same solution.
"
𝐴𝐴 − 𝐷𝐷 𝑥𝑥 + 𝐵𝐵 − 𝐸𝐸 𝑦𝑦 = 𝐶𝐶 − 𝐹𝐹
𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 = 𝐹𝐹
" (2𝐴𝐴 + 𝐷𝐷)𝑥𝑥 + (2𝐵𝐵 + 𝐸𝐸)𝑦𝑦 = 𝐶𝐶 + 2𝐹𝐹
𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 = 𝐹𝐹
" 𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵 = 𝐶𝐶
−3𝐷𝐷𝐷𝐷 − 3𝐸𝐸𝐸𝐸 = −3𝐹𝐹
"
𝐴𝐴 − 5𝐷𝐷 𝑥𝑥 + 𝐵𝐵 − 5𝐸𝐸 𝑦𝑦 = 𝐶𝐶 − 5𝐹𝐹
𝐷𝐷𝐷𝐷 + 𝐸𝐸𝐸𝐸 = 𝐹𝐹
" 𝐴𝐴𝐴𝐴 + (𝐵𝐵 + 𝐸𝐸)𝑦𝑦 = 𝐶𝐶
𝐴𝐴 + 𝐷𝐷 𝑥𝑥 + 𝐸𝐸𝐸𝐸 = 𝐶𝐶 + 𝐹𝐹
Section 3 – Video 7
Solution Sets to Inequalities with Two Variables
The senior class is raising money for Grad Bash. The students’
parents are donating cakes. The students plan to sell entire
cakes for $20.00 each and slices of cake for $3.00 each. If they
want to raise at least $500.00, how many of each could they
sell?
List some possibilities for the number of whole cakes and cake
slices students could sell to reach their goal of raising at least
$500.00.
Define the variable and write an inequality to represent the
situation.
Graph the region where the solutions to the inequality would
lie.
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Section 3: Solving Equations and Inequalities with Two Variables
What is the difference between the ordered pairs that fall on
the line and the ones that fall in the shaded area?
Each class is performing at the school-wide pep rally. The
freshman class presentation must be less than 55 minutes. They
decide to have a combination of skits that last five minutes
and dance routines that last three minutes.
Define the variables and write an inequality to represent the
situation.
What does the 𝑦𝑦-intercept represent?
Graph the region of the solutions to the inequality.
What does the 𝑥𝑥-intercept represent?
If you were senior class president at your school and wanted
to raise the most money possible, what would be your
recommendation? Why?
If you are unsure about where to shade, try a
test point in the region you shaded. Make
sure that it gives you a true statement when
you plug it into the inequality.! Section 3: Solving Equations and Inequalities with Two Variables
69
Try It!
What is the difference between the ordered pairs that fall on
the line and the ones that fall in the shaded area?
The freshman class wants to include at least 120 people in the
pep rally. Each skit will have 15 people, and the dance
routines will feature 12 people.
List some possible combinations of skits and dance routines.
Define the variables and write an inequality to represent the
situation.
What does the 𝑦𝑦-intercept represent?
Graph the region of the solutions to the inequality.
What does the 𝑥𝑥-intercept represent?
If you were freshman class president at your school, what
combination would you recommend and why?
70
Section 3: Solving Equations and Inequalities with Two Variables
BEAT THE TEST!
1. Fill in the boxes with the correct linear inequality.
𝑦𝑦 ≥ 2
!
!
!
𝑥𝑥 − 1
𝑥𝑥 − 1
𝑦𝑦 < 2(𝑥𝑥 + 1)
𝑦𝑦 > 𝑥𝑥 − 2
𝑦𝑦 < 2
!
!
𝑥𝑥 − 1
𝑦𝑦 ≤ 2(𝑥𝑥 + 1)
Number of
Basketballs
𝑦𝑦 ≤ 2
!
2. At the beginning of the school year, Coach De Leon made
a trip to a sports shop. Basketballs cost $20.00 each, and
soccer balls cost $18.00 each. He had a budget of $150.00.
The graph shown below represents the number of
basketballs and soccer balls he can buy given his budget
constraint.
Number of
Soccer Balls
Part A: Explain why Coach De Leon’s graph is correct.
Part B: Determine whether these combinations of
basketballs, 𝑏𝑏, and soccer balls, 𝑠𝑠, can be
purchased.
Yes
No
𝑏𝑏 = 5 𝑏𝑏 = 2 𝑏𝑏 = 7 𝑏𝑏 = 0 𝑏𝑏 = 8 𝑏𝑏 = 6 𝑏𝑏 = 4
𝑠𝑠 = 3 𝑠𝑠 = 4 𝑠𝑠 = 3 𝑠𝑠 = 8 𝑠𝑠 = 0 𝑠𝑠 = 3 𝑠𝑠 = 7
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Section 3: Solving Equations and Inequalities with Two Variables
!
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71
Section 3 – Video 8
Finding Solution Sets to Systems of Linear Inequalities
Juan’s parents asked him to pay for his car insurance. He
needs to earn at least $50.00 a week to cover the payments.
Because he has football practice, he can work at most eight
hours each week. Juan can earn $10.00 per hour mowing
yards and $12.00 per hour washing cars.
Define the variables and write a system of linear inequalities to
represent the situation.
Lauren is having a party and has invited 24 friends. She plans
to purchase sodas that cost $5.00 for a 12-pack and chips that
cost $3.00 per bag. She wants each friend to have at least two
sodas. The most Lauren can spend is $35.00.
Define the variables and write a system of inequalities to
represent the situation.
Graph the region where the solutions to the inequality would
lie.
Represent the solution to the system of linear inequalities on
the graph below.
Name two different solutions for Lauren’s situation.
Identify two different solutions for Juan’s situation.
72
Section 3: Solving Equations and Inequalities with Two Variables
Try It!
BEAT THE TEST!
Anna is an avid reader. Her generous grandparents gave her
money for her birthday, and she decided to spend at most
$150.00 on books. Barnes and Noble is running a special: all
paperback books are $8.00 and hardback books are $12.00.
She wants to purchase at least 12 books.
Define the variables and write a system of linear inequalities to
represent the situation.
1. Every week, Megan works at Publix during the day and at a
private security company at night. She can work a
maximum of 40 hours a week. After she calculates all her
expenses, she concludes that she needs to earn at least
$440.00 every week. Megan earns $12.00 an hour at Publix
and $10.00 an hour as a private security guard. On the
graph below, represent the possible hours Megan could
work at each place.
Graph the region where the solutions to the inequality would
lie.
Name two different solutions for Anna’s situation.
!
Section 3: Solving Equations and Inequalities with Two Variables
73
2. Tatiana is reviewing for the Algebra 1 End-of-Course exam.
She made this graph representing a system of inequalities:
Part A: Circle the ordered pairs below that represent
solutions to the system of inequalities.
(−8, 3)
(5, 5)
(−3, 8)
(1, 9)
−1, 9
(−9, 1)
−4, 9
(−2, 7)
(9, 6)
(0, 9)
(1, 6) (0, 0) Part B: Derive the system of inequalities that describes the
region of the graph Tatiana drew.
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Section 3: Solving Equations and Inequalities with Two Variables