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Solving Equations with
11-3 Variables on Both Sides
Learn to solve equations with variables on
both sides of the equal sign.
Course 3
Solving Equations with
11-3 Variables on Both Sides
Some problems produce equations that have
variables on both sides of the equal sign.
Solving an equation with variables on both
sides is similar to solving an equation with a
variable on only one side. You can add or
subtract a term containing a variable on both
sides of an equation.
Course 3
Solving Equations with
11-3 Variables on Both Sides
Additional Example 1A: Solving Equations with
Variables on Both Sides
Solve.
4x + 6 = x
4x + 6 = x
– 4x
– 4x
6 = –3x
6 = –3x
–3
–3
–2 = x
Course 3
Subtract 4x from both sides.
Divide both sides by –3.
Solving Equations with
11-3 Variables on Both Sides
Helpful Hint
Check your solution by substituting the value
back into the original equation. For example,
4(-2) + 6 = -2 or -2 = -2.
Course 3
Solving Equations with
11-3 Variables on Both Sides
Additional Example 1B: Solving Equations with
Variables on Both Sides
Solve.
9b – 6 = 5b + 18
9b – 6 = 5b + 18
– 5b
– 5b
Subtract 5b from both sides.
4b – 6 = 18
+6 +6
4b = 24
4b = 24
4
4
b=6
Course 3
Add 6 to both sides.
Divide both sides by 4.
Solving Equations with
11-3 Variables on Both Sides
Additional Example 1C: Solving Equations with
Variables on Both Sides
Solve.
9w + 3 = 9w + 7
9w + 3 = 9w + 7
– 9w
– 9w
3≠
Subtract 9w from both sides.
7
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
Course 3
Solving Equations with
11-3 Variables on Both Sides
Helpful Hint
if the variables in an equation are eliminated
and the resulting statement is false, the
equation has no solution.
Course 3
Solving Equations with
11-3 Variables on Both Sides
Check It Out: Example 1A
Solve.
5x + 8 = x
5x + 8 = x
– 5x
– 5x
8 = –4x
8 = –4x
–4
–4
–2 = x
Course 3
Subtract 5x from both sides.
Divide both sides by –4.
Solving Equations with
11-3 Variables on Both Sides
Check It Out: Example 1B
Solve.
3b – 2 = 2b + 12
3b – 2 = 2b + 12
– 2b
– 2b
Subtract 2b from both sides.
b–2=
+2
b
=
Course 3
12
+ 2 Add 2 to both sides.
14
Solving Equations with
11-3 Variables on Both Sides
Check It Out: Example 1C
Solve.
3w + 1 = 3w + 8
3w + 1 = 3w + 8
– 3w
– 3w
1≠
Subtract 3w from both sides.
8
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
Course 3
Solving Equations with
11-3 Variables on Both Sides
To solve multi-step equations with variables on
both sides, first combine like terms and clear
fractions. Then add or subtract variable terms
to both sides so that the variable occurs on
only one side of the equation. Then use
properties of equality to isolate the variable.
Course 3
Solving Equations with
11-3 Variables on Both Sides
Additional Example 2A: Solving Multi-Step Equations
with Variables on Both Sides
Solve.
10z – 15 – 4z = 8 – 2z - 15
10z – 15 – 4z = 8 – 2z – 15
6z – 15 = –2z – 7 Combine like terms.
+ 2z
+ 2z
Add 2z to both sides.
8z – 15
+ 15
8z
8z
8
z
Course 3
=
=8
= 8
8
=1
–7
+15 Add 15 to both sides.
Divide both sides by 8.
Solving Equations with
11-3 Variables on Both Sides
Additional Example 2B: Solving Multi-Step Equations
with Variables on Both Sides
y
3y 3
7
+
–
=y–
5
5
10
4
y
3y
7
+
– 3 =y–
5
5
10
4
y
3y
7
3
20 5 + 5 –
= 20 y – 10 Multiply by the LCD,
4
20.
7
y
3
3y
20 5 + 20 5
– 20 4 = 20(y) – 20 10
(
()
)
( )
(
)
()
( )
4y + 12y – 15 = 20y – 14
16y – 15 = 20y – 14
Course 3
Combine like terms.
Solving Equations with
11-3 Variables on Both Sides
Additional Example 2B Continued
16y – 15 = 20y – 14
– 16y
– 16y
–15 = 4y – 14
+ 14
–1 = 4y
–1 = 4y
4
4
–1= y
4
Course 3
+ 14
Subtract 16y from both
sides.
Add 14 to both sides.
Divide both sides by 4.
Solving Equations with
11-3 Variables on Both Sides
Check It Out: Example 2A
Solve.
12z – 12 – 4z = 6 – 2z + 32
12z – 12 – 4z = 6 – 2z + 32
8z – 12 = –2z + 38 Combine like terms.
+ 2z
+ 2z
Add 2z to both sides.
10z – 12 =
38
+ 12
+12 Add 12 to both sides.
10z = 50
10z = 50
Divide both sides by 10.
10
10
z=5
Course 3
Solving Equations with
11-3 Variables on Both Sides
Check It Out: Example 2B
y
5y 3
+
+
=y–
4
6
4
y
5y
+
+3 =y–
4
6
4
y
5y
+
+3
= 24
4
6
4
6
8
6
8
by the LCD,
(
) (
) Multiply
24.
6
y
3
5y
24(4 ) + 24( 6 )+ 24( 4)= 24(y) – 24( 8 )
24
6
y–
8
6y + 20y + 18 = 24y – 18
26y + 18 = 24y – 18
Course 3
Combine like terms.
Solving Equations with
11-3 Variables on Both Sides
Check It Out: Example 2B Continued
26y + 18 = 24y – 18
– 24y
– 24y
2y + 18 =
– 18
– 18
– 18
2y = –36
2y = –36
2
2
y = –18
Course 3
Subtract 24y from both
sides.
Subtract 18 from both
sides.
Divide both sides by 2.
Solving Equations with
11-3 Variables on Both Sides
Additional Example 4 Continued
Now find the amount of money Jamie spends each
morning.
Choose one of the original
1.25 + 2d
expressions.
1.25 + 2(0.25) = 1.75 Jamie spends $1.75 each
morning.
Find the number of doughnuts Jamie buys on Tuesday.
Let n represent the
0.25n = 1.75
number of doughnuts.
0.25n 1.75
Divide both sides by 0.25.
0.25 = 0.25
n = 7; Jamie bought 7 doughnuts on Tuesday.
Course 3
Solving Equations with
11-3 Variables
Insert Lesson
Title
Here
on Both
Sides
Lesson Quiz
Solve.
1. 4x + 16 = 2x x = –8
2. 8x – 3 = 15 + 5x x = 6
3. 2(3x + 11) = 6x + 4 no solution
1
1
x = 36
4. 4 x = 2 x – 9
5. An apple has about 30 calories more than an
orange. Five oranges have about as many calories
as 3 apples. How many calories are in each?
An orange has 45 calories. An apple
has 75 calories.
Course 3