7.2 Solve Linear Systems by Substitution
Goal  Solve systems of linear equations by substitution.
Warm-up #1:
QUESTION TO ANSWER How can you solve a system of linear equations without graphing?
STEP 1 Read the problem
The math club is selling two sizes of origami stars: large for $3 each and small for $2 each. As
of today, they have taken in $96.
STEP 2 Write an equation
Write an equation that models the situation. Let x represent the number of small stars sold
and y represent the number of large stars sold.
Equation: _______________________
STEP 3 One quantity is known
Suppose you know one quantity. How would you use this information to find the other quantity?
Use the information in the tables and the equation you wrote in Step 2 to complete the tables.
Small Sold Large Sold
0
18
30
Small Sold Large Sold
2
10
20
STEP 4 A relationship is known
Suppose you know there is a relationship between the two quantities. How would you use this
information to find the amounts sold? For example: Suppose the math club sold twice as many large
stars as small stars. Write an equation to model this situation.
Equation: _______________
STEP 5 A system of equations
Call the equation you wrote in Step 2 Equation 1 and the Equation you wrote in Step 4 Equation 2.
Combine these two equations to form a system of linear equations. (Rewrite them here).
Equation 1: __________________
Equation 2: __________________
STEP 6 Solve the system
Replace y in Equation 1 with 2x because Equation 2 gives y = 2x. Record the resulting equation:
Solve this equation for x.
x = ________
Substitute your value for x into one of the equations and solve for y.
y = ________
How many small stars did the club sell? How many large stars?
1
Large Stars: _________
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Small Stars: _________
Warm-up #2:
Watch the following video on Khan Academy. Take notes while you watch.
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-ofequations/v/solving-linear-systems-by-substitution
Example 1:
x + 2y = 9
and
3x + 5y = 20
Solution: _________ Check:
DON’T FORGET TO CHECK YOUR SOLUTION IN BOTH OF THE ORIGINAL EQUATIONS!
Example 2:
The sum of two numbers is 70. Their difference is 11. What are the numbers?
Equations:
Solution: _________ Check:
2
DON’T FORGET TO CHECK YOUR SOLUTION IN BOTH OF THE ORIGINAL EQUATIONS!
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__________________________________________________________________________________
SOLVING A LINEAR SYSTEM USING THE SUBSTITUTION METHOD
Step 1 Solve one of the equations 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 for the
original variable.
Example 3 - Write a system of equations to solve a word problem and use the
substitution method to find the solution.
Fundraising A wilderness group is selling cans of nuts and boxes of microwaveable popcorn to
raise money for a trip. A can of nuts sells for $4.50 and a box of microwaveable popcorn sells
for $3. The group sells $252 in nuts and popcorn and they sell twice as many boxes of popcorn
as cans of nuts.
a. Let x be the number of boxes of popcorn and let y be the number of cans of nuts sold.
Write an equation that relates the number of boxes of popcorn sold to the number of cans
of nuts sold.
Write an equation that gives the total amount of money made in terms of x and y.
c.
Solve the system to find out how many boxes of popcorn the group sold. How many cans
of nuts did the group sell?
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3
b.