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3-4
Solving Equations with Variables on Both Sides
Preview
Warm Up
California Standards
Lesson Presentation
3-4
Solving Equations with Variables on Both Sides
Warm Up
Solve.
1. 2x + 9x – 3x + 8 = 16
x=1
2. –4 = 6x + 22 – 4x x = –13
3. 2 + x = 5 1 x = 34
7 7
7
4. 9x – 2x = 3 1
16 4
8
x = 50
3-4
Solving Equations with Variables on Both Sides
California
Standards
Extension of
AF4.1 Solve two-step
linear equations and inequalities in one variable
over the rational numbers, interpret the solution
or solutions in the context from which they arose,
and verify the reasonableness of the results.
Also covered: AF1.1
3-4
Solving Equations with 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
To collect the variable terms on
one side, subtract 4x from both
sides.
Since x is multiplied by -3,
divide both sides by –3.
3-4
Solving Equations with Variables on Both Sides
Helpful Hint
You can always check your solution by
substituting the value back into the original
equation.
3-4
Solving Equations with 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 To collect the variable terms on one
side, subtract 5b from both sides.
4b – 6 = 18
Since 6 is subtracted from
+6 +6
4b, add 6 to both sides.
4b = 24
4b = 24
4
4
b=6
Since b is multiplied by 4,
divide both sides by 4.
3-4
Solving Equations with 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
To collect the variable terms
on one side, subtract 9w from
both sides.
3≠
7
There is no solution. There is no number that can be
substituted for the variable w to make the equation
true.
3-4
Solving Equations with 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.
3-4
Solving Equations with 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
To collect the variable terms on
one side, subtract 5x from both
sides.
Since x is multiplied by –4,
divide both sides by –4.
3-4
Solving Equations with Variables on Both Sides
Check It Out! Example 1B
Solve.
3b – 2 = 2b + 12
3b – 2 = 2b + 12
– 2b
– 2b
b–2=
+2
b
=
To collect the variable terms
on one side, subtract 2b
from both sides.
12
+ 2 Since 2 is subtracted from b,
add 2 to both sides.
14
3-4
Solving Equations with Variables on Both Sides
Check It Out! Example 1C
Solve.
3w + 1 = 3w + 8
3w + 1 = 3w + 8
– 3w
– 3w
1≠
To collect the variable terms
on one side, 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.
3-4
Solving Equations with Variables on Both Sides
To solve more complicated equations, you may
need to first simplify by combining like terms
or clearing fractions. Then add or subtract to
collect variable terms on one side of the
equation. Finally, use properties of equality to
isolate the variable.
3-4
Solving Equations with 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
=
=8
8
=
8
=1
–7
+15 Add 15 to both sides.
Divide both sides by 8.
3-4
Solving Equations with 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 Multiply by the LCD,
3
20 5 + 5 –
= 20 y – 10
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
Combine like terms.
3-4
Solving Equations with 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
+ 14
Subtract 16y from both
sides.
Add 14 to both sides.
Divide both sides by 4.
3-4
Solving Equations with 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
3-4
Solving Equations with 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
y–
6
8
6y + 20y + 18 = 24y – 18
26y + 18 = 24y – 18
Combine like terms.
3-4
Solving Equations with 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
Subtract 24y from both
sides.
Subtract 18 from both
sides.
Divide both sides by 2.
3-4
Solving Equations with Variables on Both Sides
Additional Example 3: Business Application
Daisy’s Flowers sells 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. What is the price?
Write an equation for each service. Let c represent
the total cost and r represent the number of
roses.
total cost
is
flat fee plus cost
Daisy’s:
c
=
39.95
+
Other:
c
=
26.00
+
for each rose
2.95
4.50
r
r
3-4
Solving Equations with Variables on Both Sides
Additional Example 3 Continued
Now write an equation showing that the costs are equal.
39.95 + 2.95r = 26.00 + 4.50r
– 2.95r
39.95
– 26.00
13.95
– 2.95r
=
26.00 + 1.55r
– 26.00
=
Subtract 2.95r from
both sides.
1.55r
Subtract 26.00 from
both sides.
13.95
1.55r
Divide both sides by
=
1.55
1.55
1.55.
9=r
The two bouquets from either florist would cost the
same when purchasing 9 roses.
3-4
Solving Equations with Variables on Both Sides
Additional Example 3 Continued
To find the cost, substitute 9 for r into either
equation.
Daisy’s:
c = 39.95 + 2.95r
Other florist:
c = 26.00 + 4.50r
c = 39.95 + 2.95(9)
c = 26.00 + 4.50(9)
c = 39.95 + 26.55
c = 26.00 + 40.50
c = 66.5
c = 66.5
The cost for a bouquet with 9 roses at either florist
is $66.50.
3-4
Solving Equations with Variables on Both Sides
Check It Out! Example 3
Marla’s Gift Baskets sells a muffin basket for
$22.00 plus $2.25 for every balloon. A competing
service sells a similar muffin basket for $16.00
plus $3.00 for every balloon. Find the number of
balloons that would make both baskets cost the
same price.
Write an equation for each service. Let c represent
the total cost and b represent the number of
balloons.
total cost
is
flat fee plus cost for each balloon
Marla’s:
c
=
22.00
+
2.25
Other:
c
=
16.00
+
3.00
b
b
3-4
Solving Equations with Variables on Both Sides
Check It Out! Example 3 Continued
Now write an equation showing that the costs are equal.
22.00 + 2.25b = 16.00 + 3.00b
– 2.25b
22.00
– 2.25b
Subtract 2.25b from
both sides.
= 16.00 + 0.75b
– 16.00
– 16.00
6.00
=
6.00
0.75
0.75b
Subtract 16.00 from
both sides.
0.75b
Divide both sides by
0.75
0.75.
8=b
The two services would cost the same when purchasing
a muffin basket with 8 balloons.
=
3-4
Solving Equations with Variables on Both Sides
Solve.
Lesson Quiz
1. 4x + 16 = 2x x = –8
2. 8x – 3 = 15 + 5x x = 6
3. 2(3x + 11) = 6x + 4 no solution
4. 1 x = 1 x – 9 x = 36
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.