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2-2 Solving Two-Step Equations
Vocabulary
equivalent equations
2-2 Solving Two-Step Equations
Many equations contain more than one operation,
such as 2x + 5 = 11.
This equation contains multiplication and addition.
Equations that contain two operations require two
steps to solve. Identify the operations in the
equation and the order in which they are applied to
the variable. Then use inverse operations to undo
them in reverse over one at a time.
Operations in the equation


First x is multiplied by 2.
Then 5 is added.
To solve
 Subtract 5 from both

sides of the equation.
Then divide both
sides by 2.
2-2 Solving Two-Step Equations
2x + 5 =11
–5 –5
2x = 6
Subtract 5 from both sides of
the equation.
Divide both sides of the
equation by 2.
x=3
The solution set is {3}.
Each time you perform an inverse operation, you
create an equation that is equivalent to the original
equation. Equivalent equations have the same
solutions, or the same solution set. In the example
above, 2x + 5 = 11, 2x = 6, and x = 3 are all
equivalent equations.
2-2 Solving Two-Step Equations
Additional Example 1A: Solving Two-Step Equations
Solve 18 = 4a + 10.
18 = 4a + 10
–10
8
–10
= 4a
2=a
First a is multiplied by 4. Then 10 is
added.
Subtract 10 from both sides.
8 = 4a is equivalent to 18 = 4a + 10.
Since a is multiplied by 4, divide
both sides by 4 to undo the
multiplication.
The solution set is {2}.
2-2 Solving Two-Step Equations
Additional Example 1B: Solving Two-Step Equations
Solve 5t – 2 = –32.
5t – 2 = –32
+2
5t
+2
= –30
t = –6
First t is multiplied by 5. Then 2 is
subtracted.
Add 2 to both sides.
5t = –30 is equivalent to 5t – 2 = –32.
Since t is multiplied by 5, divide
both sides by 5 to undo the
multiplication.
The solution set is {–6}.
2-2 Solving Two-Step Equations
Check It Out! Example 1a
Solve the equation. Check your answer.
–4 + 7x = 3
–4 + 7x = 3
+4
+4
7x = 7
x=1
First x is multiplied by 7. Then –4 is
added.
Add 4 to both sides.
7x = 7 is equivalent to –4 + 7x = 3.
Since x is multiplied by 7, divide
both sides by 7 to undo the
multiplication.
The solution set is {1}.
2-2 Solving Two-Step Equations
Check It Out! Example 1a Continued
Solve the equation. Check your answer.
Check –4 + 7x
=3
–4 + 7(1)
3
–4 + 7
3
3
3
To check your solution,
substitute 1 for x in the
original equation.
2-2 Solving Two-Step Equations
Check It Out! Example 1b
Solve the equation. Check your answer.
1.5 = 1.2y – 5.7
1.5 = 1.2y – 5.7 First y is multiplied by 1.2. Then 5.7
is subtracted.
+ 5.7
+5.7 Add 5.7 to both sides.
7.2 = 1.2y
6=y
7.2 = 1.2y is equivalent to
1.5 = 1.2y – 5.7.
Since y is multiplied by 1.2, divide
both sides by 1.2 to undo the
multiplication.
The solution set is {6}.
2-2 Solving Two-Step Equations
Check It Out! Example 1b Continued
Solve the equation. Check your answer.
Check 1.5 = 1.2y – 5.7
1.5 1.2(6) – 5.7
1.5
7.2 – 5.7
1.5
1.5 
To check your solution,
substitute 6 for y in the
original equation.
2-2 Solving Two-Step Equations
Check It Out! Example 1c
Solve the equation. Check your answer.
First n is divided by 7. Then 2 is added.
–2 –2
=0
Subtract 2 from each side.
= 0 is equivalent to
+ 2 = 2.
Since n is divided by 7, multiply both
sides by 7 to undo the division.
n=0
The solution set is {0}.
2-2 Solving Two-Step Equations
Check It Out! Example 1c Continued
Solve the equation. Check your answer.
Check
To check your solution,
substitute 0 for n in the
original equation.

2-2 Solving Two-Step Equations
Additional Example 2A: Solving Two-Step Equations
That Contain Fractions
Solve the equation.
Method 1 Use fraction operations.
Since is subtracted from , add
to
both sides to undo the subtraction.
2-2 Solving Two-Step Equations
Additional Example 2A Continued
Since y is divided by 8 multiply both
sides by 8.
Simplify.
y = 16
The solution set is {16}.
2-2 Solving Two-Step Equations
Additional Example 2A Continued
Method 2 Multiply by the least common denominator
(LCD) to clear fractions.
Multiply both sides by 8, the LCD of
the fractions.
Distribute 8 on the left side.
y – 6 = 10
+6 +6
y
= 16
Simplify. Since 6 is subtracted from y,
add 6 to both sides to undo the
subtraction.
The solution set is {16}.
2-2 Solving Two-Step Equations
Additional Example 2A Continued
Check your answer.
Check
To check your solution, substitute
16 for y in the original equation.

2-2 Solving Two-Step Equations
Additional Example 2B: Solving Two-Step Equations
That Contain Fractions
Solve the equation.
Method 1 Use fraction operations.
Since is added to
, subtract
from both sides to undo the addition.
2-2 Solving Two-Step Equations
Additional Example 2B Continued
Since r is multiplied by multiply both
sides by , the reciprocal.
Simplify.
The solution set is
.
2-2 Solving Two-Step Equations
Additional Example 2B Continued
Method 2 Multiply by the least common denominator
(LCD) to clear the fractions.
Multiply both sides by 12, the
LCD of the fractions.
Distribute 12 on the left side.
2-2 Solving Two-Step Equations
Additional Example 2B Continued
8r + 9 = 7
– 9 –9
8r
=–2
Simplify. Since 9 is added 8r, subtract
9 from both sides to undo the
addition.
Since r is multiplied by 8, divide both
sides 8 to undo the multiplication.
The solution set is
.
2-2 Solving Two-Step Equations
Additional Example 2B Continued
Check your answer.
Check
To check your solution, substitute
for r in the original equation.

2-2 Solving Two-Step Equations
Helpful Hint
You can multiply both sides of the equation by
any common denominator of the fractions. Using
the LCD is the most efficient.
2-2 Solving Two-Step Equations
Check It Out! Example 2a
Solve the equation. Check your answer.
Method 1 Use fraction operations.
Since is subtracted from
, add
to both sides to undo the subtraction.
2-2 Solving Two-Step Equations
Check It Out! Example 2a Continued
Since x is multiplied by multiply both
sides by , the reciprocal.
Simplify.
The solution set is
.
2-2 Solving Two-Step Equations
Check It Out! Example 2a Continued
Method 2 Multiply by the least common denominator
(LCD) to clear the fractions.
Multiply both sides by 10, the
LCD of the fractions.
Distribute 10 on the left side.
4x – 5 = 50
2-2 Solving Two-Step Equations
Check It Out! Example 2a Continued
Solve the equation. Check your answer.
4x – 5 = 50
+5
+5
4x
= 55
Simplify. Since 5 is subtracted from 4x
add 5 to both sides to undo the
subtraction.
Simplify. Since x is multiplied by 4,
divide both sides 4 to undo the
multiplication.
The solution set is
.
2-2 Solving Two-Step Equations
Check It Out! Example 2a Continued
Check
To check your solution, substitute
for x in the original equation.
5
5
2-2 Solving Two-Step Equations
Check It Out! Example 2b
Solve the equation. Check your answer.
Method 1 Use fraction operations.
Since is added to
, subtract
from both sides to undo the addition.
2-2 Solving Two-Step Equations
Check It Out! Example 2b Continued
Since u is multiplied by multiply both
sides by the reciprocal, .
Simplify.
The solution set is
.
2-2 Solving Two-Step Equations
Check It Out! Example 2b Continued
Method 2 Multiply by the least common denominator
(LCD) to clear fractions.
Multiply both sides by 8, the
LCD of the fractions.
Distribute 8 on the left side.
6u + 4 = 7
2-2 Solving Two-Step Equations
Check It Out! Example 2b Continued
Solve the equation. Check your answer.
6u + 4 = 7
–4 –4
6u
=3
Simplify. Since 4 is added to 6u
subtract 4 from both sides to undo
the addition.
Simplify. Since u is multiplied by 6,
divide both sides 6 to undo the
multiplication.
The solution set is
.
2-2 Solving Two-Step Equations
Check It Out! Example 2b Continued
Check
To check your solution, substitute
for u in the original equation.

2-2 Solving Two-Step Equations
Check It Out! Example 2c
Solve the equation. Check your answer.
Method 1 Use fraction operations.
Since is subtracted from
, add
to both sides to undo the subtraction.
2-2 Solving Two-Step Equations
Check It Out! Example 2c Continued
Since n is multiplied by multiply both
sides by the reciprocal, .
Simplify.
n = 15
The solution set is {15}.
2-2 Solving Two-Step Equations
Check It Out! Example 2c Continued
Check
To check your solution, substitute
15 for n in the original equation.

2-2 Solving Two-Step Equations
Additional Example 3: Problem-Solving Application
Jan joined the dining club at the local
café for a fee of $29.95. Being a member
entitles her to save $2.50 every time she
buys lunch. Jan calculates that she has
saved a total of $12.55 so far by joining
the club. Write and solve an equation to
find how many times Jan has eaten lunch
at the café.
1
Understand the Problem
The answer will be the number of
meals Jan had eaten.
2-2 Solving Two-Step Equations
Additional Example 3 Continued
List the important information:
• Jan paid a $29.95 dining club fee.
• Jan saved $2.50 on each meal.
• Her total savings was $12.55.
2
Make a Plan
Let m represent the number of meals that
Jan had purchased. That means that Jan
saved $2.50m. She must also subtract the
cost the dining club fee from the amount
she saved for the meals. Write an equation
to represent this situation.
2-2 Solving Two-Step Equations
Additional Example 3 Continued
initial
Savings
is
total
minus
on meals
fee
savings
2.50(m)
3
Solve
–
29.95
=
12.55
Since 29.95 is subtracted
from 2.50m, add 29.95
2.50(m) – 29.95 = 12.55
to both sides to undo
+ 29.95 +29.95
the subtraction.
2.50m
= 42.50
Since m is multiplied by
2.50, divide both sides
by 2.50 to undo the
m = 17 meals multiplication.
2-2 Solving Two-Step Equations
Additional Example 3 Continued
4
Look Back
Check that the answer is reasonable. The
savings per meal is $2.50, so if Jan had 17
meals minus the initial cost of joining the
dining club which is $29.95, it is equal to a
savings of $12.55.
2-2 Solving Two-Step Equations
Check It Out! Example 3a
Sara paid $15.95 to become a member at
a gym. She then paid a monthly
membership fee. Her total cost for 12
months was $735.95. How much was the
monthly fee?
1
Understand the Problem
The answer will be the monthly fee that
Sara had paid during the year.
2-2 Solving Two-Step Equations
Check It Out! Example 3a Continued
List the important information:
• Sara paid $15.95 for the membership.
• Sara paid a monthly fee for 12 months.
• Her total cost for 12 months was $735.95.
2
Make a Plan
Let m represent the monthly fee that Sara
paid. Sara paid that fee for 12 months. She
must also add the cost of the membership.
Write an equation to represent this
situation.
2-2 Solving Two-Step Equations
Check It Out! Example 3a Continued
Monthly
fee
12m
3
plus
initial
fee
is
total
cost.
+
15.95
=
735.95
Solve
12m + 15.95 = 735.95
–15.95 –15.95
12m
= 720.00
Since 15.95 is added to
12m, subtract 15.95
from both sides to
undo the addition.
Since m is multiplied by
12, divide both sides
by 12 to undo the
m = $60 month multiplication.
2-2 Solving Two-Step Equations
Check It Out! Example 3a Continued
4
Look Back
Check that the answer is reasonable. The
cost per month is $60.00, so if Sara paid
for 12 months plus the initial membership
fee which is $15.95, it is equal to the total
cost of $735.95.
2-2 Solving Two-Step Equations
Check It Out! Example 3b
Lynda has 12 records in her collection.
She adds the same number of new
records to her collection each month.
After 7 months Lynda has 26 records.
How many records does Lynda add each
month?
1
Understand the Problem
The answer will be the number of records
that Lynda adds each month.
2-2 Solving Two-Step Equations
Check It Out! Example 3b Continued
List the important information:
• Lynda started with 12 records.
• Lynda adds records each month.
• At the end of 7 months she had 26 records.
2
Make a Plan
Let r represent the number of records
added monthly. Lynda added new records
for 7 months to her starting number of
records 12. Write an equation to represent
this situation.
2-2 Solving Two-Step Equations
Check It Out! Example 3b Continued
Months
plus
initial
records
is
total
records
7r
+
12
=
26
3
Solve
7r + 12 = 26
–12 –12
7r
= 14
r = 2 records a month
Since 12 is added to 7r,
subtract 12 from both
sides to undo the
addition.
Since r is multiplied by 7,
divide both sides by 7
to undo the
multiplication.
2-2 Solving Two-Step Equations
Check It Out! Example 3b Continued
4
Look Back
Check that the answer is reasonable. The
number of records added per month is 2, so if
Lynda started with 12 records and added the 2
per month for 7 months, the total number is
equal to 26.