Develop Skills and Strategies
Lesson 16
Part 1: Introduction
CCSS
Solve Systems of Equations Algebraically
8.EE.C.8b
You know that solutions to systems of linear equations can be shown in graphs. Now
you will learn about other ways to find the solutions. Take a look at this problem.
Sienna wrote these equations to help solve a number riddle.
y 5 x 2 20
x 1 y 5 84
What values for x and y solve both equations?
Explore It
Use math you already know to solve the problem.
What does y 5 x 2 20 tell you about the relationship between x and y?
What does x 1 y 5 84 tell you about the relationship between x and y?
You can guess and check to solve the problem. Try 44 for x and 40 for y. Do these
numbers solve both equations?
Now try 50 for x. If x 5 50, what is y when y 5 x 2 20? Does that work with the other
equation?
Try x 5 52. What do you find?
Explain how you could find values for x and y that solve both equations.
138
L16: Solve Systems of Equations Algebraically
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Part 1: Introduction
Lesson 16
Find Out More
In Lesson 15, you learned that without actually solving, you can tell if a system of equations
will have exactly one solution, no solution, or infinitely many solutions. Here are some
examples.
x1y56
2x 1 2y 5 12
The second equation is double the first one, so they are the same
equation with the same graph and solution set. This system has
infinitely many solutions.
5x 1 y 5 3
x 5 4 2 5x
If you write both equations in slope-intercept form, you find that
y 5 25x 1 4 and y 5 25x 1 3. The lines have the same slope and
different intercepts so they are parallel. This system has no solutions.
If a system of equations has exactly one solution, like the problem on the previous page,
there are different ways you can find the solution.
You could guess and check, but that is usually not an efficient way to solve a system of
equations. You could graph each equation, but sometimes you can’t read an exact answer
from the graph. Here is one way to solve the problem algebraically.
y 5 x 2 20
x 1 y 5 84
Substitute x 2 20 for y in the second equation and solve for x.
x 1 (x 2 20) 5 84
2x 2 20 5 84
2x 5 104
x 5 52, so y 5 32
You will learn more about algebraic methods later in the lesson.
Reflect
1 How does knowing x 5 52 help you find the value of y?
L16: Solve Systems of Equations Algebraically
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139
Part 2: Modeled Instruction
Lesson 16
Read the problem below. Then explore how to use substitution to solve systems
of equations.
Solve this system of equations.
y5x12
y 1 1 5 24x
Graph It
You can graph the equations and estimate the solution.
Find the point of intersection. It looks like
(
)
the solution is close to 2 1, 11 .
2
··
2
··
5
4
3
2
1
0
2524232221
21 1 2 3 4 5
22
23
24
25
Model It
You can use substitution to solve for the first variable.
Notice that one of the equations tells you that y 5 x 1 2. This allows you to use substitution
to solve the system of equations.
Substitute x 1 2 for y in the second equation.
y5x12
y 1 1 5 24x
(x 1 2) 1 1 5 24x
Now you can solve for x.
x 1 2 1 1 5 24x
x 1 3 5 24x
3 5 25x
x 5 23
5
··
140
L16: Solve Systems of Equations Algebraically
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Part 2: Guided Instruction
Lesson 16
Connect It
Now you will solve for the second variable and analyze the solution.
2 What is the value of x? How can you find the value of y if you know the value of x?
3 Substitute the value of x in the equation y 5 x 1 2 to find the value of y.
4 Now substitute the value of x in the equation y 5 24x 2 1 to find the value of y.
5 What is the ordered pair that solves both equations? Where is this ordered pair located
on the graph?
6 Look back at Model It. How does substituting x 1 2 for y in the second equation give you
an equation that you can solve?
7 How does substitution help you to solve systems of equations?
Try It
Use what you just learned to solve these systems of equations. Show your work on a
separate sheet of paper.
8 y 2 3 5 2x
y 5 4x 2 2
9 y 5 1.4x 1 2
y 2 3.4x 5 22
L16: Solve Systems of Equations Algebraically
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141
Part 3: Modeled Instruction
Lesson 16
Read the problem below. Then explore how to solve systems of equations using
elimination.
Solve this system of equations.
2x
2 2y 5 4
3y 5 20.5x 1 2
Model It
You can use elimination to solve for one variable.
First, write both equations so that like terms are in the same position. Then try to eliminate
one of the variables, so you are left with one variable. To do this, look for a way to get
opposite coefficients for one variable in the two equations.
22y 5
x14
3y 5 20.5x 1 2
2(3y 5 20.5x 1 2)
6y 5 2x 1 4
• Multiply the second equation by
2. Now you have opposite terms:
x in the first equation and 2x in
the second equation.
22y 5 x 1 4
6y 5 2x 1 4
4y 5
8
• Add the like terms in the two
equations. The result is an
equation in just one variable.
y52
22(2) 5 x 1 4
2x 2 4 5 4
2x 5 8
x 5 28
Check:
3(2) 5 20.5(28) 1 2
65412
142
• Divide each side by 4 to solve
the equation for y.
• Substitute the value of y into
one of the original equations
and solve for x.
• Substitute your solution in the
other original equation.
L16: Solve Systems of Equations Algebraically
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Part 3: Guided Instruction
Lesson 16
Connect It
Now analyze the solution and compare methods for solving systems of equations.
10 What happens when you add opposites? Why do you want to get opposite coefficients for
one of the variables?
11 How did you get opposite coefficients for x in the solution on the previous page?
12 Why does the equation stay balanced when you add the values on each side of the
equal sign?
13 Which equation in the system was used to find the value of x?
Can you
use the other equation? Explain.
14 How is elimination like substitution? How is it different?
15 How can you check your answer?
Try It
Use what you just learned about elimination to solve this problem. Show your work on
a separate sheet of paper.
16 2x 1 y 5 9
3x 2 y 5 16
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143
Part 4: Guided Practice
Lesson 16
Study the model below. Then solve problems 17–19.
Student Model
The student divided each
term in the equation
2y 5 6x 2 2 by 2 to get an
expression equal to y.
Solve this system of equations.
3y 5 x 1 1
2y 5 6x 2 2
Look at how you could use substitution to solve a system of
equations.
2y 5 6x 2 2
2
··········
y 5 3x 2 1
Since y 5 3x 2 1, I can substitute 3x 2 1 for y in the first equation.
3(3x 2 1) 5 x 1 1
9x 2 3 5 x 1 1
8x 5 4
x51
2
··
Pair/Share
Solve the problem using
elimination.
Can it help to write both
equations in the same
form?
3y 5 1 1 1; y 5 1
2
··
2
··
1, 1
2 ··
22
Solution: 1 ··
17 What ordered pair is a solution to y 5 x 1 5 and x 2 5y 5 29?
Show your work.
Pair/Share
Discuss your solution
methods. Do you prefer
using substitution or
elimination?
144
Solution:
L16: Solve Systems of Equations Algebraically
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Part 4: Guided Practice
Lesson 16
18 Graph the equations. What is your estimate of the solution of this
system of equations?
Do the equations have the
same or different slopes?
y 5 3x 2 2
y 5 22x
Show your work.
5
4
3
2
1
0
2524232221
21 1 2 3 4 5
22
23
24
25
Solution:
19 Which of these systems of equations has no solution?
A
y5x12
4
··
y 5 4x 2 1
B
Pair/Share
Solve the problem
algebraically and
compare the solution to
your estimate.
What do you know about
the lines in a system of
equations with no
solution?
y 5 2x 2 3
3
··
y 5 2x 2 3
C
y 5 4x
y 5 4x 2 5
D
x1y53
2y 5 22x 1 6
Sheila chose D as the correct answer. How did she get that answer?
Pair/Share
Graph the solution to
verify your answer.
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145
Part 1: Introduction
Lesson 16
AT A GLANCE
Students explore a problem involving a system
of equations. They use guess-and-check to find
a solution.
Develop Skills and Strategies
Lesson 16
Part 1: Introduction
CCSS
Solve Systems of Equations Algebraically
8.EE.C.8b
You know that solutions to systems of linear equations can be shown in graphs. Now
you will learn about other ways to find the solutions. Take a look at this problem.
STEP BY STEP
Sienna wrote these equations to help solve a number riddle.
• Tell students that this page models solving a system
of equations.
y 5 x 2 20
x 1 y 5 84
What values for x and y solve both equations?
• Have students read the problem at the top of the page.
Explore It
• Work through Explore It as a class.
Use math you already know to solve the problem.
What does y 5 x 2 20 tell you about the relationship between x and y?
• When the class reaches the third bullet point, tell
students that any solution to the system has to work
for both equations, not just one.
y is 20 less than x.
What does x 1 y 5 84 tell you about the relationship between x and y?
x and y have a sum of 84.
You can guess and check to solve the problem. Try 44 for x and 40 for y. Do these
numbers solve both equations?
• Ask student pairs or groups to explain their answers
to the last three questions. Discuss what might make
certain methods difficult. For example, guess and
check can involve many trials and these equations
graph with larger numbers to seeing an exact point
might be tricky.
40 1 44 5 84, but 40 Þ 44 2 20 so the numbers don’t solve both equations.
Now try 50 for x. If x 5 50, what is y when y 5 x 2 20? Does that work with the other
equation?
If x 5 50, y 5 50 2 20, or 30. But, 50 1 30 Þ 84 so it doesn’t work in the other equation.
Try x 5 52. What do you find?
If x 5 52, then y 5 52 2 20, or 32. 52 1 32 5 84. So, the values x 5 52 and y 5 32
solve both equations.
Explain how you could find values for x and y that solve both equations.
ELL Support
Students may be unfamiliar with this usage of the
term system. Tell them that in this context it means
that a problem contains two or more equations.
You can guess and check or graph both equations to find an intersection point.
138
L16: Solve Systems of Equations Algebraically
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Mathematical Discourse
• Could you solve just one of the equations? Why or
why not?
Some students may say you cannot solve one of
the equations because you have more than one
variable. Other students may say that you can
find lots of solutions, and that any pair of x and
y that makes the equation true is a solution.
• Why would someone want to set up a system of
equations?
Students may be unsure. Some may have seen
number riddles similar to the one described in
the book. They may see how the system of
equations could help you solve the riddles.
Some may propose real-life situations in which
it is helpful or necessary to use two or more
equations to describe and solve a problem.
154
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Part 1: Introduction
Lesson 16
AT A GLANCE
Students discuss how systems of equations can have
infinite solutions, no solution, or exactly one solution.
They learn of algebraic methods that can solve systems
of equations.
STEP BY STEP
• Read Find Out More as a class.
• Point out how it is much easier to compare the two
equations if they are in the same form.
• Note that it is easy to tell when a system has infinite
solutions or no solutions. Solving for one solution,
however, takes more work.
• Ask, How did you solve for y? [substitute x 5 52 in
y 5 x 2 20]
• Ask students to share their thoughts in Reflect.
Part 1: Introduction
Lesson 16
Find Out More
In Lesson 15, you learned that without actually solving, you can tell if a system of equations
will have exactly one solution, no solution, or infinitely many solutions. Here are some
examples.
x1y56
2x 1 2y 5 12
The second equation is double the first one, so they are the same
equation with the same graph and solution set. This system has
infinitely many solutions.
5x 1 y 5 3
x 5 4 2 5x
If you write both equations in slope-intercept form, you find that
y 5 25x 1 4 and y 5 25x 1 3. The lines have the same slope and
different intercepts so they are parallel. This system has no solutions.
If a system of equations has exactly one solution, like the problem on the previous page,
there are different ways you can find the solution.
You could guess and check, but that is usually not an efficient way to solve a system of
equations. You could graph each equation, but sometimes you can’t read an exact answer
from the graph. Here is one way to solve the problem algebraically.
y 5 x 2 20
x 1 y 5 84
Substitute x 2 20 for y in the second equation and solve for x.
x 1 (x 2 20) 5 84
2x 2 20 5 84
2x 5 104
x 5 52, so y 5 32
You will learn more about algebraic methods later in the lesson.
Reflect
1 How does knowing x 5 52 help you find the value of y?
You can replace x with 52 in one of the equations to find the value of y.
SMP Tip: Students persevere in solving the
problem when they use guess and check. They
make sense of the problem when they wonder if
there might be a more efficient method (SMP 1). Tell
students to think carefully about which methods
might be best suited to each situation.
Hands-On Activity
Practice substitution using algebra tiles.
Material: a set of algebra tiles for each student pair
• Tell students they will explore this system of
equations:
2x 1 y 5 7
y 5 3x 1 2
• Have students model each equation with algebra
tiles.
• Because y equals 3x 1 2, students should replace
the y in the first equation with 3x 1 2. Have them
physically do this with the tiles and then solve the
equation for x. [x 5 1]
• Now have students model one of the equations
again, replacing the x tiles with 11 tiles. Have
them solve for y. [y 5 5] The solution is (1, 5).
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139
Real-World Connection
Equations describe various constraints on variables.
It may be that in a recipe, whether baking, or
making cement, or some other compound, you need
a certain ratio for the ingredients, and you also need
the recipe to produce a certain amount. Those are
two different constraints that can be described by
two different equations. Solving that system of
equations tells how much you need of each
ingredient to get the desired result. In another
situation, a company might source some of its
products from two different vendors. They might
use a system of equations to describe how much
total product they need and describe the different
charges they will incur from each vendor. Ask
students to think of other situations where two or
more variables might be affected by more than
one constraint.
• Have students model their own system of equations
and trade with other student pairs to solve.
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155
Part 2: Modeled Instruction
Lesson 16
AT A GLANCE
Part 2: Modeled Instruction
Students review graphing to estimate a solution
to a system of equations. Then they explore using
substitution to find an exact solution.
Lesson 16
Read the problem below. Then explore how to use substitution to solve systems
of equations.
Solve this system of equations.
STEP BY STEP
y5x12
y 1 1 5 24x
• Read the problem at the top of the page as a class.
Graph It
• In Graph It, review how to graph each equation.
You can graph the equations and estimate the solution.
5
4
3
2
1
Find the point of intersection. It looks like
• Remind students that the point of intersection is the
solution to the system of equations.
(
0
2524232221
21 1 2 3 4 5
22
23
24
25
• In Model It, discuss how both equations were solved
for y. Make sure students understand where the
equation (x 1 2) 1 1 5 24x comes from.
Model It
You can use substitution to solve for the first variable.
Notice that one of the equations tells you that y 5 x 1 2. This allows you to use substitution
to solve the system of equations.
• Use the Mathematical Discourse to discuss
alternative substitutions to method shown such as
setting both equations equal to y and setting them
equal.
Substitute x 1 2 for y in the second equation.
y5x12
y 1 1 5 24x
(x 1 2) 1 1 5 24x
Now you can solve for x.
x 1 2 1 1 5 24x
x 1 3 5 24x
SMP Tip: Students model the problem when they
use a graph to estimate the solution (SMP 4).
Remind students that the solution to a system of
equations is where the graphs of the equations
intersect.
Hands-On Activity
Review graphs of systems of equations.
Tell students to look at these four equations:
A
y5x13
B
y5x22
C
2y 5 2x 1 6
D
y 5 2x
Graph and label each equation. Ask, Will A and B
ever meet? [No. They are parallel.] Do A and B have a
solution in common? [No. There is no solution in
common.] What do you notice about A and C? (When
you solve C for y, you see they are the same
equation. Their solution set would be every ordered
pair on their graph.] What do you notice about A and
D, or B and D? [They meet at one point and would
have one solution.] Discuss how students can use
the slope of each equation to learn about the solution
set of any two equations.
156
)
the solution is close to 2 1 , 11 .
2 ··
2
··
3 5 25x
x 5 23
5
··
140
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Mathematical Discourse
• Do you think graphing is a good way to find the
solution to a system of equations? Why or why not?
Some students might think graphing is a good
way to estimate but realize it cannot accurately
find fractional answers. Some may not like
graphing and prefer an exact algebraic method.
• In the substitution method, why were the equations
solved for y? Could it have been solved differently?
Students may note that it was easy to solve
both equations for y because one equation was
already in that form. Some may suggest solving
both equations for x. Listen for responses
stating that a problem should have the same
result no matter how it is solved.
L16: Solve Systems of Equations Algebraically
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Part 2: Guided Instruction
Lesson 16
AT A GLANCE
Students revisit the problem on page 140 and discuss
how to finish solving it. They understand that the
solution to this system of equations is an ordered pair.
STEP BY STEP
• Remind students that Connect It refers to the
problem on page 140.
• Ask students what it would mean if each equation
produced a different value for y. [Getting a different
value for y would indicate that something is wrong.
When a system of linear equations has a single
solution, the solution can have just one value for y.
In such a case, it is possible that the value found for x
is incorrect.]
• Ask students why they found the value for y two
times. [to make certain the solution was true for both
equations in the system]
• As needed, review and discuss different methods of
solving equations involving fractions, such as
multiplying by a common denominator or working
with improper fractions instead of mixed numbers.
Part 2: Guided Instruction
Lesson 16
Connect It
Now you will solve for the second variable and analyze the solution.
2 What is the value of x? How can you find the value of y if you know the value of x?
x 5 23 ; I can substitute 23 for x in either equation and solve to find the value of y.
5
5
··
··
3 Substitute the value of x in the equation y 5 x 1 2 to find the value of y.
y 5 2 3 1 2; y 5 12
5
5
··
··
4 Now substitute the value of x in the equation y 5 24x 2 1 to find the value of y.
y 5 24(23) 2 1; y = 12 2 1, y = 7 or 12
5
5
5
5
··
··
··
··
5 What is the ordered pair that solves both equations? Where is this ordered pair located
(23 , 12 ) It is the point at which the lines intersect.
5 ··
5
on the graph? ··
6 Look back at Model It. How does substituting x 1 2 for y in the second equation give you
an equation that you can solve? After you substitute, the equation that results has
only one variable in it so you can solve for that variable.
7 How does substitution help you to solve systems of equations? It allows you to write
an equation with one variable instead of two and then solve for that variable.
Try It
Use what you just learned to solve these systems of equations. Show your work on a
separate sheet of paper.
8 y 2 3 5 2x
y 5 4x 2 2
(21, 8)
2
··
9 y 5 1.4x 1 2
y 2 3.4x 5 22
(2, 4.8)
141
L16: Solve Systems of Equations Algebraically
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• Help students articulate their method or thinking as
they answer each problem. Make sure they use the
appropriate algebraic terms.
TRY IT SOLUTIONS
8 Solution: 1 2 1, 8 2; Students may solve
2
··
2x 1 3 5 4x 2 2 to get x 5 5 , then solve
y 5 21 5 2 1 3 to get y 5 8.
2
··
2
··
1
ERROR ALERT: Students who wrote (2··2 , 24)
may have incorrectly solved for y 5 2x 2 3, then
found the y value from one equation. Remind
students to check their answer in both equations
to avoid mistakes.
9 Solution: (2, 4.8); Students may solve
1.4x 1 2 5 3.4x 2 2 to get x 5 2, then solve
y 5 1.4(2) 1 2 to get y 5 4.8.
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157
Part 3: Modeled Instruction
Lesson 16
AT A GLANCE
Part 3: Modeled Instruction
Students explore solving a system of equations using
the elimination method.
Read the problem below. Then explore how to solve systems of equations using
elimination.
STEP BY STEP
Solve this system of equations.
2x
• Read the problem at the top of the page as a class.
Model It
You can use elimination to solve for one variable.
First, write both equations so that like terms are in the same position. Then try to eliminate
one of the variables, so you are left with one variable. To do this, look for a way to get
opposite coefficients for one variable in the two equations.
22y 5
• Ask why multiplying the entire equation by 2 is a
legitimate step. [When you perform the same
operation on both sides of an equation, the equation
remains true.]
the equations when they determine how to
eliminate a variable (SMP 7). Tell students to write
equations in the same form to make the structure
easier to see and to help avoid mistakes.
158
x14
3y 5 20.5x 1 2
• Ask students why they multiplied the equation
3y 5 20.5x 1 2 by 2. [Multiplying the equation
by 2 made the coefficient of x equal to 21. This was a
necessary step in getting the x terms in the two
equations to cancel each other.]
SMP Tip: Students make use of the structure in
2 2y 5 4
3y 5 20.5x 1 2
• Ask students why they need to get the equations in
the same form. [Putting the equations in the same
form makes it easier to eliminate a variable and can
reduce the chances of making a mistake.]
• Ask students how they could eliminate the y terms.
[First, multiply 22y 5 x 1 4 by 3 so that the
coefficient for y becomes 26. Then, multiply
3y 5 20.5x 1 2 by 2 so that the coefficient for y
becomes 6. The y terms cancel out, leaving only the x
terms to solve for.]
Lesson 16
2(3y 5 20.5x 1 2)
6y 5 2x 1 4
• Multiply the second equation by
2. Now you have opposite terms:
x in the first equation and 2x in
the second equation.
22y 5 x 1 4
6y 5 2x 1 4
4y 5
8
• Add the like terms in the two
equations. The result is an
equation in just one variable.
• Divide each side by 4 to solve
the equation for y.
y52
22(2) 5 x 1 4
2x 2 4 5 4
2x 5 8
• Substitute the value of y into
one of the original equations
and solve for x.
x 5 28
Check:
3(2) 5 20.5(28) 1 2
65412
142
• Substitute your solution in the
other original equation.
L16: Solve Systems of Equations Algebraically
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Mathematical Discourse
• Do you prefer the substitution method or the
elimination method? Explain your reasons.
Students will give different reasons. Listen to
see if students understand that the best
method may depend on the types and numbers
of equations in the system. Substitution might
be well suited for solving one kind of problem,
and elimination might be the simpler method
for solving another.
L16: Solve Systems of Equations Algebraically
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Copying is not permitted.
Part 3: Guided Instruction
Lesson 16
AT A GLANCE
Students revisit the problem on page 142. They analyze
how to use the elimination method and why it works.
STEP BY STEP
• Point out that Connect It refers to the problem on
page 142.
• Ask students why you want to eliminate one of the
variables. [Eliminating one variable is a precondition
for finding the value of the other variable.]
• Ask students how they can check their answers.
[You could substitute the x- and y-values into both
equations to see if they make true statements.]
Part 3: Guided Instruction
Lesson 16
Connect It
Now analyze the solution and compare methods for solving systems of equations.
10 What happens when you add opposites? Why do you want to get opposite coefficients for
one of the variables? When you add opposites the sum is 0. If the coefficients are
opposites, you can add and get 0 to eliminate that variable.
11 How did you get opposite coefficients for x in the solution on the previous page?
Each term in 3y 5 20.5x 1 2 was multiplied by 2 to make the x coefficients opposites.
12 Why does the equation stay balanced when you add the values on each side of the
equal sign? The values on each side of the equal sign are equal to each other, so
you are adding the same values to both sides. Whenever you add the same value
to both sides of an equation, it stays balanced.
13 Which equation in the system was used to find the value of x? 2x 2 2y 5 4
Can you
use the other equation? Explain. Yes, it doesn’t matter which of the two equations
you use to find the value of x. You get the same value for x using both equations.
14 How is elimination like substitution? How is it different? Possible answer: With both
methods you get one equation in one of the variables. With elimination, you
Concept Extension
Solve a system of three equations.
Tell students you will determine, as a class, whether
this system of three equations has a single solution:
A
2x 1 y 5 4
B
3x 2 y 5 1
C
x 2 2y 5 2
• Tell students they will find a solution for two of
the equations and then test it on the third one.
• Point out that equations A and B are ideal for
using the elimination method to cancel out the
y-values and find a value for x. Have the class
perform the elimination method on equations
A and B to solve for x. [x 5 1]
• Have the class solve equations A and B for x 5 1.
What is the result? [y 5 2 for both equations]
Ask, Expressed as an ordered pair, what is the
solution to the system of equations A and B? [(1, 2)]
create opposite coefficients, and add. With substitution, one variable is
expressed in terms of the other and then substituted into one of the equations.
15 How can you check your answer? Substitute the values for the variables in the
other equation to be sure that you get true statements for both equations.
Try It
Use what you just learned about elimination to solve this problem. Show your work on
a separate sheet of paper.
16 2x 1 y 5 9
3x 2 y 5 16 (5, 21)
L16: Solve Systems of Equations Algebraically
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143
TRY IT SOLUTION
16 Solution: (5, 21); Students may combine the two
equations to eliminate y and solve 5x 5 25 to get
x 5 5. Substituting in x 5 5, both 2(5) 1 y 5 9 and
3(5) 2 y 5 16 simplify to y 5 21.
ERROR ALERT: Students who answered x 5 5 only
found the x-value of the solution. Remind students
that the solution to a system of equations in two
variables contains both an x-value and a y-value.
Have students substitute the x-value into either
equation to find the y-value.
• Explain that for (1, 2) to be a complete solution, it
must satisfy all three equations. As a class, test
(1, 2) in equation C. [x 2 2y 5 2; 1 2 2(2) 5 2;
23 Þ 2] Ask, So, is (1, 2) a solution of the entire
system? [no]
• Display a graph of the three-equation system for
students. Ask, What would the graph have to look
like to have a solution? [All three lines would have
to cross at one point.]
L16: Solve Systems of Equations Algebraically
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159
Part 4: Guided Practice
Part 4: Guided Practice
Lesson 16
Lesson 16
Study the model below. Then solve problems 17–19.
system of equations?
3y 5 x 1 1
2y 5 6x 2 2
Show your work.
5
4
3
2
1
2y 5 6x 2 2
2
··········
0
2524232221
21 1 2 3 4 5
22
23
24
25
y 5 3x 2 1
Since y 5 3x 2 1, I can substitute 3x 2 1 for y in the first equation.
3(3x 2 1) 5 x 1 1
9x 2 3 5 x 1 1
8x 5 4
x51
Solution: Possible answer: (0.5, 21)
2
··
Pair/Share
Can it help to write both
equations in the same
form?
Do the equations have the
same or different slopes?
y 5 3x 2 2
y 5 22x
Solve this system of equations.
Look at how you could use substitution to solve a system of
equations.
Solve the problem using
elimination.
Lesson 16
18 Graph the equations. What is your estimate of the solution of this
Student Model
The student divided each
term in the equation
2y 5 6x 2 2 by 2 to get an
expression equal to y.
Part 4: Guided Practice
3y 5 1 1 1; y 5 1
2
··
2
··
19 Which of these systems of equations has no solution?
1, 1
2 ··
22
Solution: 1 ··
A
y5x12
4
··
y 5 4x 2 1
17 What ordered pair is a solution to y 5 x 1 5 and x 2 5y 5 29?
B
Pair/Share
Solve the problem
algebraically and
compare the solution to
your estimate.
What do you know about
the lines in a system of
equations with no
solution?
y 5 2x 2 3
3
··
y 5 2x 2 3
Show your work.
Possible answer:
C
x 2 5y 5 29
y 5 4x
y 5 4x 2 5
2 5y 5 2x 2 9
Find y.
y5x15
25y 5 2x 2 9
24y 5 24
y51
Pair/Share
Discuss your solution
methods. Do you prefer
using substitution or
elimination?
144
D
Find x.
y5x15
15x15
x 5 24
x1y53
2y 5 22x 1 6
Sheila chose D as the correct answer. How did she get that answer?
Possible answer: Sheila thought that if the equations in the
system are the same, then there is no solution.
Solution:
Pair/Share
Graph the solution to
verify your answer.
(24, 1)
L16: Solve Systems of Equations Algebraically
145
L16: Solve Systems of Equations Algebraically
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AT A GLANCE
SOLUTIONS
Students use what they’ve learned to answer questions
about systems of equations.
Ex After using substitution to solve for x, the model
STEP BY STEP
17 Solution: (24, 1); Students could solve the problem
by using the substitution or elimination method.
(DOK 1)
• Ask students to solve the problems individually and
check their answers.
• When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
then uses the x-value 1 to solve for y.
2
··
18 Solution: around (0.4, 20.8); Students could solve
the problem by seeing where the lines intersect on
the graph. (DOK 1)
19 Solution: C; Sheila thought that if the equations are
identical, the system has no solution; in fact, the
system has an infinite number of solutions.
Explain to students why the other two answer
choices are not correct:
A and B are not correct because both systems have
exactly one solution. (DOK 3)
160
L16: Solve Systems of Equations Algebraically
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