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Solving Two-Step Equations
Section 2-2
Goals
Goal
• To solve two-step equations
in one variable.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• None
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.
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.
To Solve: Inverse Operations
in the Inverse Order
Ex: x + 9 = 6
5
Ask yourself:
• What is the first thing
we are doing to x?
• The second thing?
Recall the order of
operations as you
answer these questions.
• dividing by 5
• adding 9
To undo these steps, do
the inverse operations
in inverse order.
The DO-UNDO Chart
Use a chart as a shortcut to
answering the questions.
DO UNDO
÷5
-9
+9
·5
Follow the steps in the
‘undo’ column to isolate
the variable.
Ex: x + 9 = 6
5
• First subtract 9.
x+9-9=6-9
5
x = -3
5
• Then multiply by 5.
(5) x = -3(5)
5
x = -15
Example:
Complete the do-undo chart.
DO UNDO
-2
·3
÷3
+2
To solve for d:
First multiply by 3.
Then add 2.
Ex: d - 2 = 7
3
(3) d - 2 = 7(3)
3
d - 2 = 21
d - 2 = 21
+2 +2
d = 23
Example:
Remember to always use the
sign in front of the number.
DO UNDO
÷ -7
-3
+3
· -7
To solve for a:
First subtract 3.
Then multiply by -7.
Ex: 3 - a = -2
7
•
3 - a = -2
7
-3
-3
- a = -5
7
• (-7)(- a) = (-5)(-7)
7
a = 35
Your Turn:
1) 5z + 16 = 51
2) 14n - 8 = 34
3) 4b + 8 = 10
-2
The answers:
1) DO UNDO
·5
- 16
+16
÷5
2) DO UNDO
· 14
+8
-8 ÷ 14
3) DO
·4
+8
÷ -2
1) z = 7
2) n = 3
3) b = -7
UNDO
· -2
-8
÷4
Your Turn:
Solve the equation. Check your answer.
–4 + 7x = 3
–4 + 7x = 3
First x is multiplied by 7. Then –4 is added.
+4
Add 4 to both sides.
+4
7x = 7
7x = 7 is equivalent to –4 + 7x = 3.
Since x is multiplied by 7, divide both sides
by 7 to undo the multiplication.
x=1
The solution set is {1}.
Your Turn: Continued
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.
Your Turn:
Solve the equation.
1.5 = 1.2y – 5.7
1.5 = 1.2y – 5.7
+ 5.7
+5.7
7.2 = 1.2y
First y is multiplied by 1.2. Then 5.7 is
subtracted.
Add 5.7 to both sides.
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.
6=y
The solution set is {6}.
Your Turn:
Solve the equation.
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}.
Example: Two-Step
Equations with Fractions
Solve the equation.
Method 1 Use fraction operations.
Since is subtracted from , add
to
both sides to undo the subtraction.
Example: Continued
Since y is divided by 8 multiply both
sides by 8.
Simplify.
y = 16
The solution set is {16}.
Example: 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
Simplify. Since 6 is subtracted from y, add 6 to
both sides to undo the subtraction.
y
The solution set is {16}.
= 16
Example: Two-Step
Equations with Fractions
Solve the equation.
Method 1 Use fraction operations.
Since
is added to
, subtract
from both sides to undo the addition.
Example: Continued
Since r is multiplied by multiply both
sides by , the reciprocal.
Simplify.
The solution set is
.
Example: 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.
Example: 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
.
Helpful Hint
You can multiply both sides of the equation by any
common denominator of the fractions. Using the
LCD is the most efficient.
Your Turn:
Solve the equation. Check your answer.
Method 1 Use fraction operations.
Since
is subtracted from
, add
to both sides to undo the subtraction.
Your Turn: Continued
Since x is multiplied by multiply both
sides by , the reciprocal.
Simplify.
The solution set is
.
Your Turn: 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
Your Turn: Continued
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
.
Your Turn:
Solve the equation.
Method 1 Use fraction operations.
Since
is added to
, subtract
from both sides to undo the addition.
Your Turn: Continued
Since u is multiplied by multiply both
sides by the reciprocal, .
Simplify.
The solution set is
.
Your Turn: 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
Your Turn: Continued
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
.
Your Turn:
Solve the equation.
Method 1 Use fraction operations.
Since is subtracted from
, add
to both sides to undo the subtraction.
Your Turn: Continued
Since n is multiplied by
multiply both
sides by the reciprocal,
Simplify.
n = 15
The solution set is {15}.
.
Joke Time
• Who made King Arthur’s round table?
• Sir - Cumference.
• Where was the Declaration of Independence
signed?
• At the bottom.
• What was Camelot famous for?
• It’s knight life!
Assignment
• 2.2 Exercises Pg. 98 – 100: #10 – 68 even