M3: Solving Equations Notes 2.4-2.7
Page 1 of 16
Academic
Pre-Algebra
Solving Equations
Sections 2.4 to 2.7
Name_________________ Pd._____
M3: Solving Equations Notes 2.4-2.7
Sections 2.4-2.7 Vocabulary List
Rates and Unit Analysis Notes:
 rate
 unit rate
Section 2.4:
 equation
 solving an equation
 solution of an equation
Section 2.5:
 inverse operations
 equivalent equations
Page 2 of 16
M3: Solving Equations Notes 2.4-2.7
Page 3 of 16
Rates and Unit Analysis
Learning Goal: We will prepare for solving problems that involve rates and unit
analysis.
Vocabulary:
 Rate – a comparison, using division, of quantities measured in
different units
 Unit Rate – a rate whose denominator is 1 unit when the rate is
written as a fraction
Unit Rate as a Phrase
Unit Rate as a Fraction
a. 45 miles per hour
b. $3 per square foot
c. 20 feet per second
d. $1.70 per gallon
Example 1: Rates and Unit Analysis
At a grocery store, the price of bananas is $1.19 per pound. What is
the cost of 3 pounds of bananas?
ON YOUR OWN:
At a grocery store, oranges are being sold at a price of 3 for $1. What
is the cost of two dozen oranges?
M3: Solving Equations Notes 2.4-2.7
Page 4 of 16
Example 2: Rate and Variable Expressions
You fill a pool with water at a rate of 20 gallons per minute. Write an
expression for the volume of water in the pool after t minutes.
ON YOUR OWN:
While driving, Maurice averages a speed of 45 miles per hour. Write
an expression for the distance he would cover in h hours.
Example 3: Using a Formula
An ocean liner travels at a constant speed of 36 miles per hour. How
far does the ocean liner travel in 4.5 hours?
Distance  Rate  Time
d  rt
ON YOUR OWN:
A python is slithering at a constant speed of 20 feet per minute. How
far will the python travel in 0.5 hour?
M3: Solving Equations Notes 2.4-2.7
Page 5 of 16
Section 2.4: Variables and Equations
Learning Goal: We will solve equations with variables.
Vocabulary:
 Equation – a mathematical sentence formed by placing an equal
sign, =, between two expressions.
 Solution of an equation – a number that produces a true
statement when it is substituted for the variable in the equation
 Solving an equation –
Example 1: Writing Verbal Sentences as Equations
Verbal Sentence
a. The sum of x and 6 is 9.
b. The difference of 12 and y is 15.
c. The product of -4 and p is 32.
d. The quotient of n and 2 is 9.
e. The sum of b and 9 is 17.
f. The difference of z and 23 is -7.
g. The product of 12 and –k is -60.
h. The quotient of t and 5 is 11.
Equation
M3: Solving Equations Notes 2.4-2.7
Page 6 of 16
Example 2: Checking Possible Solutions
Tell whether 9 or 7 is a solution of x  5  2 .
Tell whether 52 or 60 is a solution of
x
 13 .
4
ON YOUR OWN:
Example 3: Solving Equations Using Mental Math
b. 54  h  48
a.  49  7 g
x
 12
c.
3
d.
12  y  25
M3: Solving Equations Notes 2.4-2.7
Page 7 of 16
ON YOUR OWN:
Example 4: Writing and Solving an Equation
An animal shelter charges $75 to adopt a puppy. One week they
collected $1500 in adoption fees. How many puppies were adopted
that week?
ON YOUR OWN:
EXTRA PRACTICE:
M3: Solving Equations Notes 2.4-2.7
Page 8 of 16
Section 2.5: Solving Equations Using Addition or
Subtraction
Learning Goal: We will solve one-step equations using addition or subtraction.
Vocabulary:
 Inverse operations –
 Equivalent equations –
***Solving an equation is like keeping a barbell balanced. If you add or
subtract weight from one side, you must do the same on the other side.
Example 1: Solving an Equation Using Subtraction
Solve
k  22  10 .
M3: Solving Equations Notes 2.4-2.7
Page 9 of 16
Example 2: Solving an Equation Using Addition
Solve
62  m  15 .
ON YOUR OWN:
Example 3: Writing and Solving an Equation
Rachel measures her heart rate at 123 beats per minute. This is 55
beats per minute more than her resting heart rate r. Write and solve
an equation to find Rachel’s resting heart rate.
M3: Solving Equations Notes 2.4-2.7
Page 10 of 16
A softcover book costs $17 less than its hardcover edition. The
softcover costs $5. Write and solve an equation to find the cost h of
the hardcover book.
ON YOUR OWN:
EXTRA PRACTICE:
M3: Solving Equations Notes 2.4-2.7
Page 11 of 16
Section 2.6: Solving Equations Using Multiplication or
Division
Learning Goal: We will solve one-step equations using multiplication or division.
Example 1: Solving an Equation Using Division
b. 12 w 
a.  6x  48
96
M3: Solving Equations Notes 2.4-2.7
Page 12 of 16
Example 2: Solving an Equation Using Multiplication
a. 9 
w
7
b.
a
 11
12
ON YOUR OWN:
Example 3: Writing and Solving an Equation
Molly bicycled at an average speed of 4 miles per hour to get to a post
office that was 1 mile from home. How long did it take her to reach
the post office?
ON YOUR OWN:
One of the world’s tallest office buildings is in Malaysia. The building
has 88 stories. The height of the 88 stories is 1,232 ft. What is the
height of one story?
M3: Solving Equations Notes 2.4-2.7
Page 13 of 16
Section 2.7: Decimal Operations and Equations with
Decimals
Learning Goal: We will solve equations involving decimals.
We perform operations with decimals using the same rules as those
used for integers in Chapter 1 and at the beginning of Chapter 2.
Example 1: Adding and Subtracting Decimals
b. Find the difference
a. Find the sum  2.9  (6.5) .
 25.38  (42.734)
c. Find the sum
 14.2  (17.6)
ON YOUR OWN:
d. Find the difference
4.75  (12.5)
M3: Solving Equations Notes 2.4-2.7
Page 14 of 16
Example 2: Multiplying and Dividing Decimals
Perform the indicated operation.
a.
7.2(1.5)
b.
 9.2(4.1)
c.
43.29  (4.5)
d.
 71.05  (3.5)
ON YOUR OWN:
Example 3: Solving Addition and Subtraction Equations
Solve the equation.
a. 12.6  m  9.2
b. r  2.3  1.7
M3: Solving Equations Notes 2.4-2.7
Page 15 of 16
ON YOUR OWN:
Example 4: Solving Multiplication and Division Equations
Solve the equation.
a.  12.2t  9.76
c
 10.5
b.
4
ON YOUR OWN:
M3: Solving Equations Notes 2.4-2.7
Page 16 of 16
Example 5: Writing and Solving an Equation.
The trip odometer on a car showed 229.5 miles before a salesperson
left for a sales meeting. When she reached her destination, the trip
odometer read 273.2 miles. How far did she have to drive to attend
the meeting?
ON YOUR OWN: