(8) Lesson 2.4 - Solve Equations with Variables on Each Side

1. Course 3, Lesson 2-4
Use the work backward strategy to solve Exercises 1-3.
1. Alicia arrived home at 7:45 P.M. from a restaurant. She spent 45
minutes waiting in the restaurant lobby and one and a half hours eating
dinner. If it took her 20 minutes to drive home, what time did she arrive
at the restaurant?
2. Marcus has $6 left over after his trip to the movie theater. If his movie
ticket cost $5.50 and he purchased a drink for $3.00, and a bag of
popcorn for $3.50, and a box of candy for $2.00, how much money did
he originally take to the movie theater?
3. Candace’s quiz scores are 86, 98, 85, 94, and 89. What is the minimum
score she can make on her next quiz to maintain a quiz average of at
least 90?
4. At nine months of age, a baby elephant can weigh 700 pounds. If this is
4 times the baby elephant’s birth weight, how many pounds did the
elephant weigh at birth?

2. Course 3, Lesson 2-4
ANSWERS
1. 5:10 P.M.
2. $20
3. 88
4. 175 pounds

3. WHAT is equivalence?
Expressions and Equations
Course 3, Lesson 2-4

4. • 8.EE.7
Solve linear equations in one variable.
• 8.EE.7a
Give examples of linear equations in one variable with one solution, infinitely
many solutions, or no solutions. Show which of these possibilities is the
case by successively transforming the given equation into simpler forms,
until an equivalent equation of the form x = a, a = a, or a = b results (where a
and b are different numbers).
• 8.EE.7b
Solve linear equations with rational numbers coefficients, including
equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
Course 3, Lesson 2-4 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.
Course 3, Lesson 2-4 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 solve equations with
• variables on both sides
• rational coefficients
Course 3, Lesson 2-4
Expressions and Equations

7. 1
Need Another Example?
2
3
4
Step-by-Step Example
1. Solve 8 + 4d = 5d. Check your solution.
8 + 4d = 5d Write the equation.
Write the original equation.Check 8 + 4d = 5d
8 = d
Subtraction Property of Equality
To check your solution, replace d with 8 in the original equation.
Simplify by combining like terms.
Subtract 4d from the left side of
the equation to isolate the variable.
Subtract 4d from the right side of
the equation to keep it balanced.
Replace d with 8.
The sentence is true.
8 + 4(8) = 5(8)
40 = 40
?
–4d = – 4d

8. Answer
Need Another Example?
Solve 7x + 4 = 9x. Check your solution.
2

9. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
2. Solve 6n – 1 = 4n – 5.
6n – 1 = 4n – 5 Write the equation.
Write the original equation.Check 6n – 1 = 4n – 5
Subtraction Property of Equality
Simplify.
Replace d with 8.
The sentence is true.
6(–2) – 1 = 4(–2) – 5
–13 = –13
?
2n – 1 = –5
Addition Property of Equality
Simplify.2n = –4
Mentally divide each side by 2.n = –2
– 4n = – 4n
+ 1 = + 1

10. Answer
Need Another Example?
Solve 3x – 2 = 8x + 13. Check your solution.
–3

11. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
3. Green’s Gym charges a one time fee of $50 plus $30 per session for a personal
trainer. A new fitness center charges a yearly fee of $250 plus $10 for each session
with a trainer. For how many sessions is the cost of the two plans the same?
20s = 200
Simplify.
Check Green’s Gym: $50 plus 10 sessions at $30 per session
Simplify.
Addition Property of Equality
50 + 10 • 30 = 50 + 300
250 + 10 • 10 = 250 + 100
50 + 20s = 250
Subtraction Property of Equality
Write the equation.50 + 30s = 250 + 10s
fee of $50 plus
$30 per session
Words
Let s represent the number of sessions.Variable
50 + 30s = 250 + 10sEquation
is the
same as
a fee of $250 plus
$10 per session
Division Property of Equality
s = 10
So, the cost is the same for 10 personal trainer sessions.
Simplify.
= $350
new fitness center: $250 plus 10 sessions at $10 per session
= $350
– 10s = – 10s
– 50 = – 50

12. Answer
Need Another Example?
The measure of an angle is 8 degrees more than its
complement. If x represents the measure of the angle and
90 – x represents the measure of its complement, what is
the measure of the angle?
49°

13. 1
Need Another Example?
2
3
4
Step-by-Step Example
4. Solve x – 1 = 9 – x.
x = 12
Simplify.
Multiplicative Inverse Property
Addition Property of Equality
The common denominator of the coefficients is 6.
Rewrite the equation.
Addition Property of Equality
Simplify.
Simplify.
+ 1 = + 1

14. Answer
Need Another Example?
Solve x + 2 = 7 + x.
12

15. How did what you learned
today help you answer the
WHAT is equivalence?
Course 3, Lesson 2-4
Expressions and Equations

16. How did what you learned
today help you answer the
WHAT is equivalence?
Course 3, Lesson 2-4
Expressions and Equations
Sample answers:
• When variables are on both sides of the equation, use
the properties of equality to write an equivalent
equation with the variables on one side.
• You can check the solution to an equation by replacing
the variable with the solution in the original equation.

17. Name the procedures you
would use to solve an
equation with variable terms
on both sides such as
3x + 5 = 6x + 2.
Course 3, Lesson 2-4
Ratios and Proportional RelationshipsExpressions and Equations