Solving Equations with Variables on Both Sides

1
Solving Equations with Variables on
Both Sides

• 10-3

Pre-Algebra
2
Warm Up Solve. 1. 2x 9x 3x 8 16 2. 4
6x 22 4x 3. 5 4. 3
x 1
x -13
x 34
x 50
3
Learn to solve equations with variables on both
sides of the equal sign.
4
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.
5
Example Solving Equations with Variables on Both
Sides
Solve. A. 4x 6 x
4x 6 x
4x 4x
Subtract 4x from both sides.
6 3x
Divide both sides by 3.
2 x
6
Example Solving Equations with Variables on Both
Sides
Solve. B. 9b 6 5b 18
9b 6 5b 18
5b 5b
Subtract 5b from both sides.
4b 6 18
6 6
Add 6 to both sides.
4b 24
Divide both sides by 4.
b 6
7
Example Solving Equations with Variables on Both
Sides
Solve. C. 9w 3 5w 7 4w
9w 3 5w 7 4w
3 ? 7
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
8
Try This
Solve. A. 5x 8 x
5x 8 x
5x 5x
Subtract 4x from both sides.
8 4x
Divide both sides by 4.
2 x
9
Try This
Solve. B. 3b 2 2b 12
3b 2 2b 12
2b 2b
Subtract 2b from both sides.
b 2 12
2 2
Add 2 to both sides.
b 14
10
Try This
Solve. C. 3w 1 10w 8 7w
3w 1 10w 8 7w
1 ? 8
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
11
To solve multistep 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.
12
Example Solving Multistep Equations with
Variables on Both Sides
Solve. A. 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 7
15 15
Add 15 to both sides.
8z 8
Divide both sides by 8.
z 1
13
Example Solving Multistep Equations with
Variables on Both Sides
B.
y
Multiply by the LCD.
4y 12y 15 20y 14
16y 15 20y 14
Combine like terms.
14
Example Continued
16y 15 20y 14
16y 16y
Subtract 16y from both sides.
15 4y 14
14 14
Add 14 to both sides.
1 4y
Divide both sides by 4.
15
Try This
Solve. A. 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
Divide both sides by 10.
z 5
16
Try This
B.
y
Multiply by the LCD.
6y 20y 18 24y 18
26y 18 24y 18
Combine like terms.
17
Try This Continued
26y 18 24y 18
24y 24y
Subtract 24y from both sides.
2y 18 18
18 18
Subtract 18 from both sides.
2y 36
Divide both sides by 2.
y 18
18
Example Consumer Application
Jamie spends the same amount of money each
morning. On Sunday, he bought a newspaper for
1.25 and also bought two doughnuts. On Monday,
he bought a newspaper for fifty cents and bought
five doughnuts. On Tuesday, he spent the same
amount of money and bought just doughnuts. How
many doughnuts did he buy on Tuesday?
19
Example Continued
First solve for the price of one doughnut.
Let d represent the price of one doughnut.
1.25 2d 0.50 5d
2d 2d
Subtract 2d from both sides.
1.25 0.50 3d
Subtract 0.50 from both sides.
0.50 0.50
0.75 3d
Divide both sides by 3.
The price of one doughnut is 0.25.
0.25 d
20
Example Continued
Now find the amount of money Jamie spends each
morning.
Choose one of the original expressions.
1.25 2d
Jamie spends 1.75 each morning.
1.25 2(0.25) 1.75
Find the number of doughnuts Jamie buys on
Tuesday.
Let n represent the number of doughnuts.
0.25n 1.75
Divide both sides by 0.25.
n 7 Jamie bought 7 doughnuts on Tuesday.
21
Try This
Helene walks the same distance every day. On
Tuesdays and Thursdays, she walks 2 laps on the
track, and then walks 4 miles. On Mondays,
Wednesdays, and Fridays, she walks 4 laps on the
track and then walks 2 miles. On Saturdays, she
just walks laps. How many laps does she walk on
Saturdays?
22
Try This Continued
First solve for distance around the track.
Let x represent the distance around the track.
2x 4 4x 2
2x 2x
Subtract 2x from both sides.
4 2x 2
2 2
Subtract 2 from both sides.
2 2x
Divide both sides by 2.
The track is 1 mile around.
1 x
23
Try This Continued
Now find the total distance Helene walks each day.
Choose one of the original expressions.
2x 4
Helene walks 6 miles each day.
2(1) 4 6
Find the number of laps Helene walks on
Saturdays.
Let n represent the number of 1-mile laps.
1n 6
n 6
Helene walks 6 laps on Saturdays.
24
Lesson Quiz
Solve. 1. 4x 16 2x 2. 8x 3 15 5x 3.
2(3x 11) 6x 4 4. x 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?
x 8
x 6
no solution
x 36
An orange has 45 calories. An apple has 75
calories.