IM UNIT 2 Equations and Inequalities Teacher Guide

Point out that there is nothing wrong about adding the equations in the last system. It
simply doesn't get us anywhere closer to the solution and is therefore unproductive.

Support for English Language Learners

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help
students improve their writing by providing them with multiple opportunities to clarify
their explanations through conversation. Give students time to meet with 1–2
partners to share their response to the final question about Diego’s method. Provide
listeners with prompts for feedback that will help their partner add detail to
strengthen and clarify their ideas. For example, students can ask their partner: “How
did you use Diego’s method to solve this problem?” or “Can you say more about....?”
Give students 2–3 minutes to revise their initial draft based on feedback from their
peers. This will help students produce a written generalization for how they can
determine whether adding equations is a useful strategy for solving a linear system.
Design Principle(s): Support sense-making; Optimize output (for explanation)

14.3 Adding and Subtracting Equations to
Solve Systems

15 minutes
Earlier, students saw that adding or subtracting the equations in a system creates a third
equation that can help us solve the system. In this activity, students reconnect the idea of
the solution to a system to the intersection of the graphs of the equations. They graph
each original pair of equations and the equation that results from adding or subtracting
them. They then observe that the graph of the third equation intersects the other two
graphs at the exact same point—at the intersection of the first two.

At this point, students simply get a graphical confirmation that adding or subtracting
equations can help them find the solution to a system. They are not yet expected to be
able to articulate why this is the case. That understanding will be developed over a few
upcoming lessons.

Addressing

• HSA-REI.C.6

Building Towards

• HSA-REI.C.5

Algebra1 Unit 2 Lesson 14 CC BY 2019 by Illustrative Mathematics 9

Instructional Routines

• Graph It

Launch

Remind students that earlier they added or subtracted pairs of equations to form new
equations. Explain that they will now graph each pair of equations in the systems given
earlier, as well as the third equation that came from adding or subtracting those
equations, and then make some observations about them.

Arrange students into groups of 3 and provide access to graphing technology. Assign one
system for each group member to graph. Ask students to discuss their observations after
graphing.

If possible, make available graphing technology that allows users to enter linear equations
in standard form, such as Desmos (available under Math Tools). Otherwise, give students
time to rearrange the equations into a form that can be used with the technology and to
check their equivalent equations. If time is limited, provide these equivalent equations:

System A System B System C

Support for Students with Disabilities

Action and Expression: Internalize Executive Functions. To support development of
organizational skills, check in with students within the first 2–3 minutes of work time.
Look for students who are able to find the sum or difference of the equations with
ease. If students struggle, direct students to use parentheses around the entirety of
the equation being added or subtracted to enhance clarity and reduce errors.
Supports accessibility for: Memory; Organization

Anticipated Misconceptions
When solving system B, some students may not notice that the -variable in one equation
has a positive coefficient and the other has a negative coefficient, and consequently decide
to subtract the second equation from the first, rather than to add the two equations. They
may struggle to figure out why the solution pair they find doesn't match what is on the

Algebra1 Unit 2 Lesson 14 CC BY 2019 by Illustrative Mathematics 10

graph. Suggest that they express the second equation in terms of addition, ,
and try eliminating one variable again.

Student Task Statement
Here are three systems of equations you saw earlier.

System A System B System C

For each system:

1. Use graphing technology to graph the original two equations in the system.
Then, identify the coordinates of the solution.

2. Find the sum or difference of the two original equations that would enable the
system to be solved.

3. Graph the third equation on the same coordinate plane. Make an observation
about the graph.

Student Response
System A:

1. or , or .
2. Sample response:

Algebra1 Unit 2 Lesson 14 CC BY 2019 by Illustrative Mathematics 11

3. Sample observations: All three lines intersect at . If students’ graph , the
, the
graph of the new equation is a horizontal line. If students graph

graph of the new equation is a new line between the two original lines.

System B:

1. . If students graph , the

2. or

3. Sample observations: All three lines intersect at
graph of the new equation is a vertical line.

System C:

1.

2. Sample response: or or , or (if the equations are added)
.

3. Sample observations: All three lines intersect at . If students graph ,
the graph of the new equation is a horizontal line.

Are You Ready for More?

Mai wonders what would happen if we multiply equations. That is, we multiply the
expressions on the left side of the two equations and set them equal to the
expressions on the right side of the two equations.

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1. In system B write out an equation that you would get if you multiply the two
equations in this manner.

2. Does your original solution still work in this new equation?

3. Use graphing technology to graph this new equation on the same coordinate
plane. Why is this approach not particularly helpful?

Student Response or equivalent.
1.

2. Yes

3. Sample response: The new equation is not a line and therefore not so easy to work
with.

Activity Synthesis

Display the graphs that students generated for all to see and ask students to share their
observations. Highlight that the graph of the new equation intersects the graphs of the
equations in the original system at the same point.

Time permitting, ask students to subtract the equations they previously added (or add the
equations they previously subtracted) and then to graph the resulting equation on the
same coordinate plane. Ask them to comment on the graphs. Students are likely to see
that the graphs of the new equations are no longer horizontal or vertical lines, but they
still intersect at the same point as the original graphs.

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Invite students to make some conjectures as to why the graph of the new
equation intersects the other two graphs at the same point. Without confirming or
correcting their conjectures, tell students that they will investigate this question in the
coming activities.

Lesson Synthesis

Now that students have an additional strategy for solving systems in their toolkit,
invite students to reflect on three systems seen in the synthesis of a previous lesson, in
which they made a case for solving one by substitution.

System 1 System 2 System 3

Ask students to discuss the following questions with a partner and be prepared to report
their partner's responses:

• "Look at a system that you would have chosen to solve by substitution. Would you

still choose to solve it by substitution now? Why or why not?"

• "Look at a system that you would not have chosen to solve by substitution. Would it

help to solve by elimination? Why or why not?"

With a little bit of rearranging (of the equations), all of these systems could be solved by
substitution or elimination. Students should at least recognize that systems 1 and 3 can be
efficiently solved by elimination, while system 2 can be efficiently solved by substitution.

14.4 What to Do with This System?

Cool Down: 5 minutes
Graphing technology should not be used in this cool-down.

Addressing

• HSA-REI.C.6

Algebra1 Unit 2 Lesson 14 CC BY 2019 by Illustrative Mathematics 14

Student Task Statement

Here is a system of linear equations:

1. Which would be a more helpful for solving the system: adding the two
equations or subtracting one from the other? Explain your reasoning.

2. Solve the system without graphing. Show your reasoning.

Student Response

1. Adding the equations. Sample explanation: so adding the equations

would eliminate the -variable and enable solving for .

2. (or equivalent) and . Sample reasoning:

Student Lesson Summary

Another way to solve systems of equations algebraically is by elimination. Just like
in substitution, the idea is to eliminate one variable so that we can solve for the
other. This is done by adding or subtracting equations in the system. Let’s look at an
example.

Algebra1 Unit 2 Lesson 14 CC BY 2019 by Illustrative Mathematics 15

Notice that one equation has and the other has .

If we add the second equation to the first, the and
add up to 0, which eliminates the -variable,

allowing us to solve for .

Now that we know , we can substitute 10 for in either of the equations and
find :

In this system, the coefficient of in the first equation happens to be the opposite
of the coefficient of in the second equation. The sum of the terms with -variables
is 0.

What if the equations don't have opposite coefficients
for the same variable, like in the following system?

Notice that both equations have and if we subtract

the second equation from the first, the variable will be

eliminated because is 0.

Substituting 5 for in one of the equations gives us :

Adding or subtracting the equations in a system creates a new equation. How do
we know the new equation shares a solution with the original system?

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If we graph the original equations in the system and the new equation, we can see
that all three lines intersect at the same point, but why do they?

In future lessons, we will investigate why this strategy works.

Glossary
• elimination

Lesson 14 Practice Problems

1. Problem 1
Statement

Which equation is the result of adding these two
equations?

A.
B.
C.
D.

Solution

C

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2. Problem 2
Statement

Which equation is the result of subtracting the
second equation from the first?

A.
B.
C.
D.

Solution

D

3. Problem 3
Statement

Solve this system of equations without graphing:

Solution

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4. Problem 4

Statement

Here is a system of linear equations:

Would you rather use subtraction or addition to solve the system? Explain
your reasoning.

Solution

Sample response: I would rather use addition. Adding the two equations immediately
eliminates and allows us to solve for .

5. Problem 5

Statement

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Kiran sells full boxes and half-boxes of fruit to raise money for a band trip.
He earns $5 for each full box and $2 for each half-box of fruit he sells and
earns a total of $100 toward the cost of his band trip. The equation

describes this relationship.

Solve the equation for .

Solution

Sample response:

(From Unit 2, Lesson 8.)

6. Problem 6

Statement

Match each equation with the corresponding equation solved for .
A. 1.
B. 2.
C.
3.
D.
4.
E.
5.

Solution

7. A: 5
8. B: 1
9. C: 4

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10. D: 3
11. E: 2

(From Unit 2, Lesson 8.)

12. Problem 7

Statement

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The volume of a cylinder is represented by the formula .
Find each missing height and show your reasoning.

volume (cubic inches) radius (inches) height (inches)
4
2.5

Solution

6, 5, (heights in inches)

(From Unit 2, Lesson 9.)

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13. Problem 8

Statement

Match each equation with the slope and -intercept of its graph.

A. , 1.

B. , 2.

C. , 3.

D. , 4.
E. , 5.
F. , 6.

Solution

14. A: 6
15. B: 2
16. C: 5
17. D: 3
18. E: 4
19. F: 1

(From Unit 2, Lesson 11.)

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20. Problem 9
Statement

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Solve each system of equations.
a.

b.

Solution

a.
b.

(From Unit 2, Lesson 13.)

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21. Problem 10

Statement

Elena and Kiran are playing a board game. After one round, Elena says, "You
earned so many more points than I did. If you earned 5 more points, your
score would be twice mine!"
Kiran says, "Oh, I don't think I did that much better. I only scored 9 points
higher than you did."

a. Write a system of equations to represent each student's comment. Be
sure to specify what your variables represent.

b. If both students were correct, how many points did each student score?
Show your reasoning.

Solution

a. Let represent Kiran's score and represent Elena’s score:

b. Kiran’s score is 23 and Elena’s score is 14. Sample reasoning: Substituting

for in the second equation gives or , so .

Kiran's score is 9 points higher, so it's 23.

(From Unit 2, Lesson 13.)

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Lesson 15: Solving Systems by Elimination
(Part 2)

Goals
• Explain (orally and in writing) why adding or subtracting two equations that share a

solution results in a new equation that also shares the same solution.

• Practice solving systems of linear equations by adding or subtracting equations to

eliminate a variable.

• Use a context to make sense of an equation that is the sum of two equations in a

system, and to reason about why this equation shares a solution with the system.

Learning Targets
• I can explain why adding or subtracting two equations that share a solution results in

a new equation that also shares the same solution.

Lesson Narrative

In this lesson, students continue to develop their understanding of solving systems by
elimination. Students are given a system that represents the quantities and constraints in
a situation. They interpret, in context, the solutions to the individual equations and to the
system. They then use the context to make sense of the sum of the two equations and why
it shares a solution with the equations in the given system. Along the way, students begin
to formulate a logical explanation as to why adding (or subtracting) the two equations in a
system can be helpful for identifying the solution to the system (MP3).

Students also practice solving systems by adding and subtracting equations and checking
their solutions. They also encounter systems where one variable cannot be easily
eliminated (given what they know at this point), motivating the need for another strategy.

Alignments
Building On

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with

graphs), focusing on pairs of linear equations in two variables.

Addressing

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with

graphs), focusing on pairs of linear equations in two variables.

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 1

Building Towards

• HSA-REI.C.5: 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.

Instructional Routines

• Graph It
• MLR3: Clarify, Critique, Correct
• MLR6: Three Reads
• Think Pair Share

Required Materials
Graphing technology
Examples of graphing technology are: a handheld graphing calculator, a computer with a
graphing calculator application installed, and an internet-enabled device with access to a
site like desmos.com/calculator or geogebra.org/graphing. For students using the digital
materials, a separate graphing calculator tool isn't necessary; interactive applets are
embedded throughout, and a graphing calculator tool is accessible on the student digital
toolkit page.

Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is
ideal if each student has their own device. (Desmos is available under Math Tools.)

Student Learning Goals

• Let’s think about why adding and subtracting equations works for solving

systems of linear equations.

15.1 Is It Still True?

Warm Up: 5 minutes
In this warm-up, students reason about whether and when the sums of equations are
true. The work here prepares students for the next activity, where they begin to think
about why the values that simultaneously satisfy two equations in a system also satisfy the
the equation that is a sum of those two equations.

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 2

Building Towards

• HSA-REI.C.5

Student Task Statement .
Here is an equation:

1. Perform each of the following operations and answer these questions: What
does each resulting equation look like? Is it still a true equation?

a. Add 12 to each side of the equation.

b. Add to the left side of the equation and 12 to the right side.

c. Add the equation to the equation .

2. Write a new equation that, when added to , gives a sum that is
also a true equation.

3. Write a new equation that, when added to , gives a sum that is a
false equation.

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 3

Student Response , or (or equivalent). Yes, it is still a true equation.
1. a.

b. , or (or equivalent). Yes, it is still a true
equation.

c. , or , or (or equivalent). Yes, it is
still a true equation.

2. Sample responses: , or
, or
◦ . Added to the original equation gives:

or

◦ . Added to the original equation gives:

3. Sample responses: , or
, or
◦ . Added to the original equation gives:

, which is false.

◦ . Added to the original equation gives:

, which is false.

Activity Synthesis

Ask students to share their responses to the first three questions. Discuss questions such
as:

• "Why do you think the resulting equations remain true even after we add numbers or

expressions that look different to each side?" (The numbers being added to the two

sides are always equal amounts, even though they are written in a different form.)

• "Suppose we subtract 12 from the left side of and subtract from

the right side. Would the resulting equation still be true?" (Yes. is 39

and is also 39. The same amount, 12, is subtracted from both sides.)

• "Here is another equation: . Suppose we subtract 12 from the left side of

that equation and from the right side. Would the resulting equation still be

true? Why or why not?" (No. The given equation is a false statement. Subtracting the

same amount from both sides will keep it false.)

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 4

Invite students to share their equations for the last two questions. Display the equations
for all to see. If no one shares an equation that uses a variable, give an example or two (as
shown in the Student Responses).

Make sure students understand that adding (or subtracting) the same amount to each side
of a true equation keeps the two sides equal, resulting in an equation that is also true.
Adding (or subtracting) different amounts to each side of a true equation, however, makes
the two sides unequal and thus produces a false equation.

15.2 Classroom Supplies

20 minutes
This activity serves two goals. The first goal is to further build students' intuition about the
new variable equation that comes from adding two variable equations in a system, and
about the solution to that new equation. This is done by grounding the addition and the
sum in a familiar context.

The second goal is to support students in reasoning about why the new equation shares a
solution with the original system. The context gives students a concrete mental reference,
which can be helpful for interpreting the intersection of the graphs of all three equations
and for thinking about the solution that all three equations share.

In this activity, students also encounter a system that they can solve by graphing and by
substitution, but cannot be easily solved simply by adding or subtracting the equations.
This observation may pique students' curiosity and make them wonder if it is possible to
solve all systems by elimination.

As students work, identify students who solve the system by graphing and those who solve
by substitution. Ask them to share their work later.

Addressing

• HSA-REI.C.6

Building Towards

• HSA-REI.C.5

Instructional Routines

• Graph It
• MLR6: Three Reads

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 5

• Think Pair Share

Launch
Arrange students in groups of 2 and provide access to graphing technology.

Read the opening paragraphs with the class. Invite students to explain what each equation
represents in this situation. Then, give students 2 minutes of quiet time to think about the
first set of questions, and a few minutes to discuss their thinking with their partner. Follow
with a whole-class discussion.

Ask students to share their interpretation of the solutions to each equation and how many
solutions are possible for each equation. Make sure students recognize that the solutions
are pairs of - and -values that make each equation true, and that each equation can
have many solutions because there are many possible prices of calculators and measuring
tapes that can make each equation true.

Before students proceed to the rest of the activity, ask: “If we are solving the system, what
are we really looking for?” Be sure students see that to solve the system is to find a pair of
unit prices that make the equations for both purchases true.

Support for English Language Learners

Reading, Listening, Conversing: MLR6 Three Reads. Ask students to keep their books or
devices closed and display only the image with the opening task statement, without
revealing the questions that follow. Use the first read to orient students to the
situation by asking students to describe what the situation is about without using
numbers (a teacher buys calculators and measuring tapes for her class). Use the
second read to identify quantities and relationships; ask students: “What can be
counted or measured?” Listen for, and amplify, the important quantities that vary in
relation to each other in this situation: number of calculators purchased (for each
order), number of measuring tapes purchased (for each order), and total cost of each
order. After the third read, reveal the questions that follow and invite students to
brainstorm possible strategies to find the solution to the system. This routine helps
students interpret the language within a given situation needed to create an equation.
Design Principle(s): Support sense-making

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 6

Support for Students with Disabilities

Representation: Internalize Comprehension. Provide appropriate reading
accommodations and supports to ensure student access to written directions, word
problems and other text-based content. While reading the situation aloud, explicitly
link the word problem to the corresponding algebraic expression. Point to both the
sentence and its corresponding equation while reading. Pause in between sentences
and reread or repeat gestures if necessary for increased comprehension.
Supports accessibility for: Language; Conceptual processing

Student Task Statement
A teacher purchased 20 calculators and 10 measuring tapes for her class and paid
$495. Later, she realized that she didn’t order enough supplies. She placed another
order of 8 of the same calculators and 1 more of the same measuring tape and paid
$178.50.
This system represents the constraints in this situation:

1. Discuss with a partner:
a. In this situation, what do the solutions to the first equation mean?
b. What do the solutions to the second equation mean?
c. For each equation, how many possible solutions are there? Explain how
you know.
d. In this situation, what does the solution to the system mean?

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2. Find the solution to the system. Explain or show your reasoning.

3. To be reimbursed for the cost of the supplies, the teacher recorded: “Items
purchased: 28 calculators and 11 measuring tapes. Amount: $673.50.”
a. Write an equation to represent the relationship between the numbers of
calculators and measuring tapes, the prices of those supplies, and the
total amount spent.

b. How is this equation related to the first two equations?

c. In this situation, what do the solutions of this equation mean?

d. How many possible solutions does this equation have? How many
solutions make sense in this situation? Explain your reasoning.

Student Response

1. Sample response:

a. The solutions to are all possible combinations of unit prices

for the two items that make a total of $495 when buying 20 calculators and 10

measuring tapes.

b. The solution to are all possible combinations of unit prices for

the two items that make a total of $185 when buying 8 calculators and 2

measuring tapes.

c. Many possible solutions. For example, in the first equation, and
.
would make the equation true, as would and

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 8

d. The price for a calculator and the price for a measuring tape that would make
both equations true (or satisfy both constraints).

2. and . (Students may solve by graphing or by substitution.)

3. a.

b. It is the sum of the first two equations.

c. Sample response: The solutions are all possible combinations of prices for a
calculator and for a measuring tape that make a total of $680 when 28
calculators and 11 measuring tapes are bought.

d. Sample response: It has many possible solutions, but only one solution makes
sense. The prices of a calculator and of a measuring tape in this equation are
the same as those in the other two equations. The new equation simply shows
the total numbers of supplies and the total amount paid.

Activity Synthesis
Select previously identified students to share their solution and strategy for solving the
system. Display their work (especially student-generated graphs) for all to see, or consider
diplaying this graph:

Emphasize that 21.50 and 6.50 are the unit prices of the two supplies that make both
equations true.

Next, discuss students' responses to the last set of questions. Ask questions such as:

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 9

• "The first equation was . We added to the left side and

178.50 to the right side to get . Are the two sides of the resulting

equation still equal? Why or why not?" (Yes. We’re adding equal amounts to each

side, so the two sides are still equal. The cost of buying 8 calculators and 1 measuring

tape is $178.50.)

• "Predict what the graphs of all three equations would look like." (They would intersect

at a common point.) Consider graphing all three equations for all to see.

• "In this situation, why does it make sense for the graphs intersect at the same point?

What does that point mean?" (The prices of a calculator and of a measuring tape in
the first two orders are also the prices in the combined order. The unit prices of the
items didn’t change, so it makes sense that they work for all three equations.)

• "In general, why does it make sense for a third equation that is the sum of the two

equations in a system to share a common solution with the other two?" (Adding the
two equations means adding equal values to each side of one equation, which keeps
the two sides of the sum equal. If there is a pair of and values that make the first
two equations true, that pair also makes the third equation true. )

• "Were you able to solve the system by adding or subtracting the equations, without

graphing? Why or why not?" (No, because no variables get eliminated when we add

or subtract.)

15.3 A Bunch of Systems

10 minutes
In this activity, students practice using algebra to solve systems of linear equations in two
variables and checking their solutions. Students do not have to use elimination, but the
equations in the first three systems conveniently have opposites for the coefficients of one
variable, so one variable can be easily eliminated.

In the last system, none of the coefficients of the variables are opposites. Some students
may choose to solve the system by substitution, but the process would be pretty
cumbersome. The complication students encounter here motivates the need for another
move, which students will explore in the next lesson.

Students who opt to use technology to check their solutions practice choosing tools
strategically (MP5).

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Addressing

• HSA-REI.C.6

Instructional Routines

• MLR3: Clarify, Critique, Correct

Launch
Keep students in groups of 2 and provide continued access to graphing technology, in case
needed for checking solutions.
If time is limited, ask one partner in each group to solve the first two of the systems and
the other partner to solve the last two, and then check each other's solutions.

Student Task Statement
Solve each system of equations without graphing and show your reasoning. Then,
check your solutions.
AB

CD

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Student Response

•:
•:
•:
•:

Are You Ready for More?
This system has three equations:

1. Add the first two equations to get a new equation.

2. Add the second two equations to get a new equation.

3. Solve the system of your two new equations.

4. What is the solution to the original system of equations?

Student Response
1.
2.
3.
4.

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 12

Activity Synthesis

Invite students to share their solutions and strategies for the first three systems and how
they check their solutions. Then, focus the discussion on the last system. Solicit the
strategies students used for approaching that system. If someone solved it by substitution,
display the work for all to see. If no one did, ask if it is possible to do. (If time permits,
consider asking students to attempt to do so, or demonstrating that strategy to illustrate
that it is not exactly efficient.)

Discuss questions such as:

• “What happens if we add or subtract the equations?” (No variables are eliminated, so

it doesn’t help with solving.)

• “Why were you able to solve the first three systems by elimination but not the last

one?” “What features were in the equations in those systems but not in the last one?”
(In the first three systems, at least one variable in each pair of equations have the
same or opposite coefficients, so when the terms were added or subtracted, the
result is 0.)

We need new moves! Tell students that they will explore another way to solve a system by
elimination in an upcoming lesson.

Support for English Language Learners

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their

solutions and strategies, present an incorrect method of solving a linear system. For

example, “After the first step when solving the system and by

elimination, the resulting equation is .” Ask students to identify the error,

critique the reasoning, and write a correct explanation. As students discuss with a

partner, listen for students who clarify the meaning of the solution to a system. Invite

students to share their critiques and corrected explanations with the class. Listen for

and amplify the language students use to describe what happens when adding two

equations together and explain what the solution should be. This helps students

evaluate and improve on the written mathematical arguments of others, as they

clarify how to find the solution of a linear system by using the elimination method.

Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 13

Lesson Synthesis

To highlight the idea that adding equal amounts to the two sides of a true equation keeps
the equation true, display this system for all to see:

Ask students if there is a pair of - and -values that make both equations true. Once

students suggest as the solution, use graphing technology to graph the equations

and verify the solution. Display the graph for all to see.

Survey the class to see who have solved the system by adding the left side of each

equation ( and ) and then adding the right side (7 and 20). (Most or all

students are likely to have done this.) Then, discuss with students:

• "Why is it OK to add an expression, , to one side of the first equation, but add a

number, 7, to the other side?" (Because and 7 are equal)

• "What is the new equation? ( ) Does the pair still make this equation

true?" (Yes. Graph this equation to illustrate that it intersects the original graphs at

the same point.)

• "Suppose we add 7 to the left side of the first equation and to the right side.

What is the new equation?" ( ) "Would the same pair of

and still work?" (Yes. Display the graph of the new equation so students could see

that the new line intersects at the same point. Students could also verify by

substitution.)

• "Suppose that, instead of adding, we subtract 7 from the left side and subtract

from the right side. What is the new equation?" ( ). "Would

the same pair of and still work?" (Yes. Graph the new equation to show that this

line, too, intersects at the same point. Or students could verify by substitution.)

• "Why does the new equation still give the same solution even though we didn't add

the numbers or expressions from the same side of each equation?" (If the two sides
of an equation are equal, it doesn't matter which side of one equation is added to
which side of another equation. The new equation will stay balanced.)

15.4 Putting New Equations to Work

Cool Down: 5 minutes

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 14

Building On

• HSA-REI.C.6

Addressing

• HSA-REI.C.6

Building Towards

• HSA-REI.C.5

Student Task Statement
On a family outing, Tyler bought 5 cups of hot cocoa and 4 pretzels for $18.40.
Some of his family members would like a second serving, so he went back to the
same food stand and bought another 2 cups of hot cocoa and 4 pretzels for $11.20.

Here is a system of equations that represent the quantities and constraints in this
situation.

1. What does the solution to the system, , represent in this situation?

2. If we add the second equation to the first equation, we have a new equation:
.

Explain why the same pair that is a solution to the two original equations

is also a solution to this new equation.

3. Does the equation help us solve the original system? If you

think so, explain how it helps. If you don't think so, explain why not and what

would help us solve the system.

Stu1d. ent Response

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 15

The values of and represent the price of a cup of hot cocoa and the price of a
pretzel that Tyler paid for.

2. Sample explanations:

◦ The new equation shows the total number of cups of hot cocoa and total

number of pretzels on one side, and the total amount Tyler spent on the other
side. The price of each cup of hot cocoa and the price of each pretzel haven't
changed.

◦ and 11.20 are equal, so adding to one side of an equation and

adding 11.20 to the other side means adding equal amounts to the two sides of
an equation. This keeps the two sides equal. The same values of and that
make the original equations true haven't changed.

3. No, it doesn't help. Sample reasoning: It doesn't eliminate a variable. What would

help is subtracting the second equation from the first to get . Then we can

find and solve for .

Student Lesson Summary

When solving a system with two equations, why is it acceptable to add the two
equations, or to subtract one equation from the other?

Remember that an equation is a statement that says two things are equal. For

example, the equation says a number has the same value as another

number . The equation says that has the same value as 12.

If and are true statements, then adding to and adding
to means adding the same amount to each side of . The result,

, is also a true statement.

As long as we add an equal amount to each side of a true equation, the two sides of
the resulting equation will remain equal.

We can reason the same way about adding variable
equations in a system like this:

In each equation, if is a solution, the expression on the left of the equal sign

and the number on the right are equal. Because is equal to -1:

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 16

• Adding to and adding -1 to 17 means

adding an equal amount to each side of

. The two sides of the new equation,

, stay equal.

The - and -values that make the original
equations true also make this equation true.

• Subtracting from and subtracting -1

from 17 means subtracting an equal amount from

each side of . The two sides of the new

equation, , stay equal.

The -variable is eliminated, but the -value that
makes both the original equations true also makes
this equation true.

From , we know that . Because 6 is also the -value that makes the

original equations true, we can substitute it into one of the equations and find the

-value.

The solution to the system is , or the point on the graphs

representing the system. If we substitute 6 and 11 for and in any of the

equations, we will find true equations. (Try it!)

Lesson 15 Practice Problems

1. Problem 1

Statement

Solve this system of linear equations without graphing:

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 17

Solution

Sample response: . I added the 2 equations and solved for and

got 3.6. I substituted it back into the first equation to solve for and got -2.5.

2. Problem 2

Statement

Select all the equations that share a solution with this system of equations.

A.
B.
C.
D.
E.

Solution

["B", "E"]

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 18

3. Problem 3

Statement

Students performed in a play on a Friday and a Saturday. For both
performances, adult tickets cost dollars each and student tickets cost
dollars each.

On Friday, they sold 125 adult tickets and 65 student tickets, and collected
$1,200. On Saturday, they sold 140 adult tickets and 50 student tickets, and
collect $1,230.

This situation is represented by this system of equations:

a. What could the equation mean in this situation?

b. The solution to the original system is the pair and . Explain

why it makes sense that this pair of values is also the solution to the

equation .

Solution

Sample response:

a. could mean that when 265 adult tickets and 115 student
tickets were sold, $2,430 was collected.

b. The equation is the sum of the two original equations.

One side of the equation shows the total earnings in terms of the number of

adult tickets and student tickets sold from both days. The other side shows the

total amount of money collected from both days. The pair and is a

solution to this equation because the cost of each type of ticket is still the same.

4. Problem 4

Statement

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 19

Which statement explains why shares a solution with
this system of equations:

A. Because is the product of the two equations in the

system of equations, it the must share a solution with the system of

equations.

B. The three equations all have the same slope but different -intercepts.
Equations with the same slope but different -intercepts always share a
solution.

C. Because is equal to 29, I can add to the left side of

and add 29 to the right side of the same equation. Adding

equivalent expressions to each side of an equation does not change the

solution to the equation.

D. Because is equal to 55, I can subtract from the left

side of and subtract 55 from its right side. Subtracting

equivalent expressions from each side of an equation does not change

the solution to the equation.

Solution

D

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 20

5. Problem 5

Statement

Select all equations that can result from adding
these two equations or subtracting one from the
other.

A.
B.
C.
D.
E.
F.

Solution

["B", "C", "E"]

(From Unit 2, Lesson 14.)

6. Problem 6

Statement

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 21

Solve each system of equations.
a.

b.

Solution

a.
b.

(From Unit 2, Lesson 13.)

7. Problem 7

Statement

Here is a system of equations:

Would you rather use subtraction or addition to solve the system? Explain
your reasoning.

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 22

Solution

Sample responses:

◦ I would rather use addition because this eliminates and allows us to solve for

.

◦ I would rather use subtraction because this eliminates and allows us to solve

for .

(From Unit 2, Lesson 14.)

8. Problem 8

Statement

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 23

The box plot represents the distribution of the number of free throws that
20 students made out of 10 attempts.

After reviewing the data, the value recorded as 1 is determined to have been
an error. The box plot represents the distribution of the same data set, but
with the minimum, 1, removed.

The median is 6 free throws for both plots.
a. Explain why the median remains the same when 1 was removed from
the data set.

b. When 1 is removed from the data set, does the mean remain the same?
Explain your reasoning.

Solution

a. The median remains the same because removing an extreme value from a data
set tends not to have much effect or no effect on the median. In this case, there
may be multiple people who made 6 free throws.

b. Sample response: Since the mean depends on all the values in a data set, the
removal of one value should affect the mean. In this case, the mean increases
because 1 is smaller than the mean of the data set.

(From Unit 1, Lesson 10.)

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 24

9. Problem 9

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 25

Statement

In places where there are crickets, the outdoor temperature can be predicted

by the rate at which crickets chirp. One equation that models the relationship

between chirps and outdoor temperature is , where is the

number of chirps per minute and is the temperature in degrees Fahrenheit.

a. Suppose 110 chirps are heard in a minute. According to this model, what
is the outdoor temperature?

b. If it is outside, about how many chirps can we expect to hear in

one minute?

c. The equation is only a good model of the relationship when the outdoor

temperature is at least . (Below that temperature, crickets aren't

around or inclined to chirp.) How many chirps can we expect to hear in a

minute at that temperature?

d. On the coordinate plane, draw a graph that represents the relationship
between the number of chirps and the temperature.

e. Explain what the coefficient in the equation tells us about the
relationship.

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 26

f. Explain what the 40 in the equation tells us about the relationship.

Solution

a.

b. About 140 chirps

c. About 60 chirps

d. See graph.

e. Sample response: The coefficient is the slope of
the graph. It tells us that every additional chirp
represents a increase in temperature.

f. Sample response: The point tells us that

when there are no chirps, the outdoor temperature

is about . However, because the equation is

only a good model of the relationship when the

temperature is at least or the number of

chirps is at least 60, any point to the left of

on the graph doesn't really reflect what's

happening in the situation.

(From Unit 2, Lesson 10.)

Algebra1 Unit 2 Lesson 15 CC BY 2019 by Illustrative Mathematics 27

Lesson 16: Solving Systems by Elimination
(Part 3)

Goals
• Recognize that multiplying an equation by a factor creates an equivalent equation

whose graph is the same as that of the original equation.

• Solve systems of equations by multiplying one or both equations by a factor and then

adding or subtracting the equations to eliminate a variable.

• Understand that solving a system by elimination or by substitution entails creating

one or more equivalent systems that would enable us to solve the original one.

Learning Targets
• I can solve systems of equations by multiplying each side of one or both equations by

a factor, then adding or subtracting the equations to eliminate a variable.

• I understand that multiplying each side of an equation by a factor creates an

equivalent equation whose graph and solutions are the same as that of the original
equation.

Lesson Narrative

This is the last lesson in a series of three lessons on solving systems of equations by
elimination. Two new ideas are introduced here.

The first idea is that we can multiply one or both equations in the system by a factor to
make it possible to eliminate a variable. Prior to this point, students worked only with
systems where at least one variable in the equations had the same coefficient or with
opposite coefficients, making the variable removable when the equations were added or
subtracted.

Here students see that this is not a requirement for a system to be solvable by elimination.
We can first multiply one or both equations by a factor—chosen strategically so that the
coefficients of one variable become equal or opposites. Then, the variable can be
eliminated by adding an original equation and the new equation, or by subtracting one
from the other.

The second new idea is that, whenever we multiply equations in a system by a factor, add
or subtract the equations, or otherwise manipulate the equations, we are essentially

Algebra1 Unit 2 Lesson 16 CC BY 2019 by Illustrative Mathematics 1

creating an equivalent system that would help us get closer to finding the solution of the
original system.

The work here builds on earlier work on acceptable moves and equivalent equations.
Students should understand and be able to explain two key threads:

• Multiplying two equal things by the same value results in two things that are also

equal. Variable values that make the original equation true also make the new
equation true.

• Adding one equation in a system to another equation is an example of adding an

equal amount to each side of an equation. The two sides of the resulting equation
are still equal. If the original equations in the system share a set of variable values
that make them true, the new equation also shares this set of values.

As they work to process and articulate these key ideas, students practice constructing
logical arguments (MP3).

Alignments
Building On

• HSA-REI.A.1: Explain each step in solving a simple equation as following from the

equality of numbers asserted at the previous step, starting from the assumption that
the original equation has a solution. Construct a viable argument to justify a solution
method.

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with

graphs), focusing on pairs of linear equations in two variables.

Addressing

• HSA-REI.C.5: 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.

• HSA-REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with

graphs), focusing on pairs of linear equations in two variables.

Instructional Routines

• Graph It

• MLR8: Discussion Supports

• Take Turns

Algebra1 Unit 2 Lesson 16 CC BY 2019 by Illustrative Mathematics 2

• Think Pair Share

Required Materials
Graphing technology
Examples of graphing technology are: a handheld graphing calculator, a computer with a
graphing calculator application installed, and an internet-enabled device with access to a
site like desmos.com/calculator or geogebra.org/graphing. For students using the digital
materials, a separate graphing calculator tool isn't necessary; interactive applets are
embedded throughout, and a graphing calculator tool is accessible on the student digital
toolkit page.

Pre-printed slips, cut from copies of the blackline master

Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is
ideal if each student has their own device. (Desmos is available under Math Tools.)

Be prepared to display a graph using technology for all to see.

Student Learning Goals

• Let's find out how multiplying equations by a factor can help us solve systems

of linear equations.

16.1 Multiplying Equations By a Number

Warm Up: 10 minutes
In this warm-up, experiment with the graphical effects of multiplying both sides of
an equation in two variables by a factor.

Students are prompted to multiply one equation in a system by several factors to generate
several equivalent equations. They then graph these equations on the same coordinate
plane that shows the graphs of the original system. Students notice that no new graphs
appear on the coordinate plane and reason about why this might be the case.

The work here reminds students that equations that are equivalent have all the same
solutions, so their graphs are also identical. Later, students will rely on this insight to
explain why we can multiply one equation in a system by a factor—which produces an
equivalent equation—and solve a new system containing that equation instead.

Algebra1 Unit 2 Lesson 16 CC BY 2019 by Illustrative Mathematics 3

Building On

• HSA-REI.A.1

Addressing

• HSA-REI.C.5

Instructional Routines

• Graph It

Launch

Display the equation for all to see. Ask students:

• "What is the solution to this equation?" ( )

• "If we multiply both sides of the equation by a factor, say, 4, what equation would we

have?" ( ) "What is the solution to this equation?" ( )

• "What if we multiply both sides by 100?" ( ) "By 0.5?"

()

• "Is the solution to each of these equations still ?" (Yes)

Remind students that these equations are one-variable equations, and that multiplying
both sides of a one-variable equation by the same factor produces an equivalent equation
with the same solution.

Ask students: "What if we multiply both sides of a two-variable equation by the same
factor? Would the resulting equation have the same solutions as the original equation?"
Tell students that they will now investigate this question by graphing.

Arrange students in groups of 2–4 and provide access to graphing technology to each
group. To save time, consider asking group members to divide up the tasks. (For example,
one person could be in charge of graphing while the others write equivalent equations,
and everyone analyze the graphs together.)

Student Task Statement
Consider two equations in a system:

Algebra1 Unit 2 Lesson 16 CC BY 2019 by Illustrative Mathematics 4

1. Use graphing technology to graph the equations. Then, identify the
coordinates of the solution.

2. Write a few equations that are equivalent to equation A by multiplying both
sides of it by the same number, for example, 2, -5, or . Let’s call the resulting
equations A1, A2, and A3. Record your equations here:

a. Equation A1:

b. Equation A2:

c. Equation A3:

3. Graph the equations you generated. Make a couple of observations about the
graphs.

Student Response

1.

2. Sample response:
a. Equation A1:

b. Equation A2:

c. Equation A3:

3. Sample response: No new graphs appeared. The graphs of the equivalent equations

are identical to that of Equation A. The graphs of Equations A1, A2, and A3 all

intersect the graph of Equation B at .

Algebra1 Unit 2 Lesson 16 CC BY 2019 by Illustrative Mathematics 5