1. The document presents three systems of equations related to word problems involving Justin and Karin's allowance, costs for recreation programs, and ages of Dewan and Adrianne. 2. Graphs are helpful for solving systems of equations because they allow you to visually check solutions obtained through algebraic methods. Graphing the equations shows their point of intersection, which indicates the solution. 3. Learning to solve systems of equations algebraically and using graphs helps answer word problems leading to two linear equations in two variables, as required by the Common Core standards.

1. Course 3, Lesson 3-8
1. Last year, Justin and his sister, Karin, earned a total of $468 in
allowance. If Justin earned $52 more than Karin in allowance,
write a system of equations that represents their allowances.
2. Mrs. Kung spent the same amount on two programs at the local
recreation center. The aerobics class costs an initial fee of $10
plus $3 per class. The pottery class costs an initial fee of $6 plus
$5 per class. Write a system of equations to represent the cost
for the two programs.
3. The sum of Dewan’s age and three times Adrianne’s age is 32.
The difference between Dewan’s age and Adrianne’s age is 4.
Write a system of equations that can be used to find Dewan’s
age and Adrianne’s age.

2. Course 3, Lesson 3-8
ANSWERS
1. j + k = 468
j = 52 + k
2. 10 + 3x = y
6 + 5x = y
3. d + 3a = 32
d – a = 4

3. WHY are graphs helpful?
Expressions and Equations
Course 3, Lesson 3-8

4. • 8.EE.8
Analyze and solve pairs of simultaneous linear equations.
• 8.EE.8b
Solve systems of two linear equations in two variables
algebraically, and estimate solutions by graphing the equations.
Solve simple case by inspection.
• 8.EE.8c
Solve real-world and mathematical problems leading to two linear
equations in two variables.
Course 3, Lesson 3-8 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
Expressions and Equations

5. Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
7 Look for and make use of structure.
Course 3, Lesson 3-8 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
Expressions and Equations

6. To
• write and solve a system of
equations by substitution
Course 3, Lesson 3-8
Expressions and Equations

7. • substitution
Course 3, Lesson 3-8
Expressions and Equations

8. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
1. Solve the system of equations algebraically.
y = x – 3
y = 2x
Since y is equal to 2x, you can replace y with 2x in the first equation.
y = x – 3
Graph the system.Check
Since x = –3 and y = 2x, then y = –6 when x = –3.
The solution of this system of equations is (–3, –6).
7
Write the equation.
2x = x – 3 Replace y with 2x.
Subtraction Property of Equality
x = –3 Simplify.
–x = –x

9. Answer
Need Another Example?
Solve the system y = x + 15 and y = 4x
algebraically.
(5, 20)

10. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
2. Solve the system of equations algebraically.
y = 3x + 8
8x + 4y = 12
Since x = –1, replace x with –1 in the equation
y = 3x + 8 to find the value of y.
The solution of this system is (–1, 5).
Write the equation.
8x + 4(3x + 8) = 12 Replace y with 3x + 8.
Collect like terms.20x + 32 = 12
Subtraction Property of Equality
8x + 4y = 12
Distributive Property
8x + 12x + 32 = 12 Simplify.
8x + 4 • 3x + 4 • 8 = 12
Simplify.
Division Property of Equality
Simplify.
20x = –20
x = –1
y = 3(–1) + 8 or 5
y = 3x + 8
–32 = –32

11. Answer
Need Another Example?
Solve the system y = 4x – 3 and 3x + 2y = 38
algebraically.
(4, 13)

12. 1
Need Another Example?
2
3
Step-by-Step Example
3. A total of 75 cookies and cakes were donated for a bake sale to raise
money for the football team. There were four times as many cookies
donated as cakes.
Write a system of equations to represent this situation.
Draw a bar diagram. Then write the system.
y = 4x There were 4 times as many cookies donated as cakes.
x + y = 75 The total number of cakes and cookies is 75.

13. Answer
Need Another Example?
A store sold 84 black and gray T-shirts one
weekend. They sold 5 times as many black
T-shirts as gray T-shirts. Write a system of
equations to represent this situation.
Sample answer: b + g = 84; b = 5g

14. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
4. A total of 75 cookies and cakes were donated for a bake sale
to raise money for the football team. There were four times as
many cookies donated as cakes. The situation can be
represented by y = 4x and x + y = 75. Solve the system
algebraically. Interpret the solution.
Since y is equal to 4x, you can replace y with 4x.
x = 15 Simplify.
Since x = 15 and y = 4x, then y = 60 when x = 15.
The solution is (15, 60). This means that 15 cakes
and 60 cookies were donated.
x + y = 75 Write the equation.
x + 4x = 75 Replace y with 4x.
5x = 75 Simplify.
Division Property of Equality

15. Answer
Need Another Example?
A store sold 84 black and gray T-shirts one
weekend. They sold 5 times as many black
T-shirts as gray T-shirts. The situation can be
represented by b + g = 84 and b = 5g.
Solve the system algebraically. Interpret the solution.
(70, 14); The store sold 70 black and
14 gray T-shirts.

16. How did what you learned
today help you answer the
WHY are graphs helpful?
Course 3, Lesson 3-8
Expressions and Equations

17. How did what you learned
today help you answer the
WHY are graphs helpful?
Course 3, Lesson 3-8
Expressions and Equations
Sample answer:
• You can use a graph to check the solution to a system
of equations that was solved algebraically.

18. Solve the system of
equations and
x = y – 6
algebraically.
Ratios and Proportional RelationshipsExpressions and Equations
x y 
1
15
2
Course 3, Lesson 3-8