Solving Equations with
7-3 Variables on Both Sides
Learn to solve equations with variables on
both sides of the equal sign.
Solving Equations with
7-3 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.
Solving Equations with
7-3 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
Subtract 4x from both sides.
Divide both sides by –3.
Solving Equations with
7-3 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.
Solving Equations with
7-3 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
Add 6 to both sides.
Divide both sides by 4.
Solving Equations with
7-3 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.
Solving Equations with
7-3 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.
Solving Equations with
7-3 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
Subtract 5x from both sides.
Divide both sides by –4.
Solving Equations with
7-3 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
=
12
+ 2 Add 2 to both sides.
14
Solving Equations with
7-3 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.
Solving Equations with
7-3 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.
Solving Equations with
7-3 Variables on Both Sides
Additional Example 2: 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
=
=8
= 8
8
=1
–7
+15 Add 15 to both sides.
Divide both sides by 8.
Solving Equations with
7-3 Variables on Both Sides
Check It Out: Example 2
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
Solving Equations with
7-3 Variables on Both Sides
Additional Example 3: Business Application
Daisy’s Flowers sell a rose bouquet for
$39.95 plus $2.95 for every rose. A
competing florist sells a similar bouquet
for $26.00 plus $4.50 for every rose. Find
the number of roses that would make both
florists’ bouquets cost the same price.
Solving Equations with
7-3 Variables on Both Sides
Additional Example 3 Continued
39.95 + 2.95r = 26.00 + 4.50r
– 2.95r
39.95
– 2.95r
=
– 26.00
13.95
Subtract 2.95r from
both sides.
26.00 + 1.55r
Subtract 26.00 from
both sides.
– 26.00
=
Let r represent the
price of one rose.
1.55r
13.95
1.55r
Divide both sides by
=
1.55
1.55
1.55.
9=r
The two services would cost the same when
purchasing 9 roses.
Solving Equations with
7-3 Variables on Both Sides
Additional Example 5: Solving Literal Equations for a
Variable
The equation t = m + 10e gives the test
score t for a student who answers m
multiple-choice questions and e essay
questions correctly. Solve this equation
for e.
t = m + 10e
Locate e in the equation.
t = m + 10e
Since m is added to 10e,
–m –m
subtract m from both sides.
t–m=
10e
Since e is multiplied 10, divide
t – m = 10e
both sides by 10.
10
10
t–m= e
10
Solving Equations with
7-3 Variables on Both Sides
Lesson Quiz
Solve.
1. 4x + 16 = 2x
x = –8
2. 8x – 3 = 15 + 5x
x=6
no solution
3. 2(3x + 11) = 6x + 4
x = 36
1
1
4. x = x – 9
4
2
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.