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Section 3-2:
Solving Linear
Systems by
Substitution
Solving a Linear System by Substitution
 Step 1: Solve one equation for one of its
variables.
 Step 2: Substitute the expression from step 1 into
the other equation and solve for the other
variable.
 Step 3: Substitute the value from Step 2 into the
revised equation from Step 1 and solve.
 Step 4: Check the solution in each of the original
equations.
Example 1
Use Substitution
Solve the system using substitution.
y = 2x
4x – y = 6
Equation 1
Equation 2
SOLUTION
Substitute 2x for y in Equation 2. Solve for x.
4x – y = 6
4x – 2x = 6
2x = 6
x=3
Write Equation 2.
Substitute 2x for y.
Combine like terms.
Solve for x.
Example 1
Use Substitution
Substitute 3 for x in Equation 1. Solve for y.
y = 2x
Write Equation 1.
y = 2( 3)
Substitute 3 for x.
y=6
Solve for y.
You can check your answer by substituting 3 for x and 6
for y in both equations.
ANSWER
The solution is ( 3, 6).
Example 2
Use Substitution
Solve the system using substitution.
3x + 2y = 7
x – 2y = – 3
Equation 1
Equation 2
SOLUTION
STEP 1 Solve Equation 2 for x.
x – 2y = – 3
x = 2y – 3
Choose Equation 2 because the
coefficient of x is 1.
Solve for x to get revised
Equation 2.
Example 2
Use Substitution
STEP 2 Substitute 2y – 3 for x in Equation 1. Solve for y.
3x + 2y = 7
3( 2y – 3) + 2y = 7
6y – 9 + 2y = 7
8y – 9 = 7
8y = 16
y=2
Write Equation 1.
Substitute 2y – 3 for x.
Use the distributive property.
Combine like terms.
Add 9 to each side.
Solve for y.
Example 2
Use Substitution
STEP 3 Substitute 2 for y in revised Equation 2. Solve
for x.
x = 2y – 3
Write revised Equation 2.
x = 2( 2 ) – 3
Substitute 2 for y.
x=1
Simplify.
STEP 4 Check by substituting 1 for x and 2 for y in the
original equations.
Equation 1
3x + 2y = 7
Equation 2
Write original equations.
x – 2y = – 3
Example 2
Use Substitution
?
3( 1 ) + 2( 2 ) = 7
Substitute for x and y. 1 – 2( 2 ) = – 3
?
3+4=7
?
1 – 4 = –3
7=7
ANSWER
The solution is ( 1, 2).
?
Simplify.
Solution checks.
–3 = –3
Checkpoint
Use Substitution
Solve the system using substitution. Tell which equation
you chose to solve and use for the substitution. Explain.
1. 2x + y = 3
3x + y = 0
ANSWER
(– 3, 9 ). Sample answer: The second equation; this
equation had 0 on one side and the coefficient of y
was 1, so I solved for y to obtain y = – 3x.
Checkpoint
Use Substitution
Solve the system using substitution. Tell which equation
you chose to solve and use for the substitution. Explain.
2. 2x + 3y = 4
x + 2y = 1
ANSWER
( 5, –2). Sample answer: The second equation; the
coefficient of x in this equation was 1, so solving for x
gave a result that did not involve any fractions.
Checkpoint
Use Substitution
Solve the system using substitution. Tell which equation
you chose to solve and use for the substitution. Explain.
3.
3x – y = 5
4x + 2y = 10
ANSWER
( 2, 1). Sample answer: The first equation; the coefficient
of y in this equation was –1, so solving for y gave a
result that did not involve any fractions.
Homework:
p. 135 17-22 all
Example 3
Write and Use a Linear System
Museum Admissions On one day, the Henry Ford
Museum in Dearborn, Michigan, admitted 4400 adults
and students and collected $57,200 in ticket sales.
The price of admission is $14 for an adult and $10 for a
student. How many adults and how many students were
admitted to the museum that day?
SOLUTION
VERBAL
MODEL
Number
of adults
+
Number
of students
=
Total number
admitted
Adult Number Student
Number
Total amount
•
•
=
+
price of adults
price
of students
collected
Example 3
LABELS
ALGEBRAIC
MODEL
Write and Use a Linear System
Number of adults = x
(adults)
Number of students = y
(students)
Total number admitted = 4400
(people)
Price for one adult = 14
(dollars)
Price for one student = 10
(dollars)
Total amount collected = 57,200
(dollars)
x + y = 4400
Equation 1 (number admitted)
14x + 10y = 57,200 Equation 2 (amount collected)
Example 3
Write and Use a Linear System
Use substitution to solve the linear system.
x = 4400 – y
Solve Equation 1 for x; revised
Equation 1.
14 (4400 – y) + 10y = 57,200
Substitute 4400 – y for x in
Equation 2.
61,600 – 14y + 10y = 57,200
Use the distributive property.
61,600 – 4y = 57,200
– 4y = – 4400
y = 1100
Combine like terms.
Subtract 61,600 from each
side.
Divide each side by – 4.
Example 3
Write and Use a Linear System
x = 4400 – y
Write revised Equation 1.
x = 4400 – 1100
Substitute 1100 for y.
x = 3300
Simplify.
ANSWER
There were 3300 adults and 1100 students admitted to
the Henry Ford Museum that day.
Checkpoint
Write and Use a Linear System
4. On another day, the Henry Ford Museum admitted
1300 more adults than students and collected
$56,000. How many adults and how many students
were admitted to the museum that day?
ANSWER
2875 adults and 1575 students