1. Warm Up
Solve.
1. 2x + 9x – 3x + 8 = 16
2. –4 = 6x + 22 – 4x
x = 1
x = –13

2. Learn to solve equations with variables on both sides of the equal sign.

3. Move variables to the left!
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.
Copy in Notes!
Move variables to the left!

4. Example 1: Solving Equations with Variables on Both Sides
Solve.
4x + 6 = x
4x + 6 = x
– x – x
Subtract x from both sides.
3x + 6 = 0 –6
Subtract 6 from both sides.
3x = -6 3 3
Divide both sides by 3
x = -2

5. Check your solution by substituting the value back into the original equation. For example, 4(-2) + 6 = -2 or -2 = -2.
Helpful Hint

6. Example 2: 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
Add 6 to both sides.
4b = 24 4b 4 24 =
Divide both sides by 4.
b = 6

7. Example 3: Solving Equations with Variables on Both Sides
Solve.
9w + 3 = 9w + 7
9w + 3 = 9w + 7
– 9w – 9w
Subtract 9w from both sides.
3 ≠
No solution. There is no number that can be substituted for the variable w to make the equation true.

8. If the variables in an equation are eliminated and the resulting statement is false, the equation has no solution.
Helpful Hint

9. Example 4
Solve.
5x + 8 = x
5x + 8 = x
– x – x
Subtract x from both sides.
4x +8 = 0
Subtract 8 from both sides.
–8 -8 4x 4 –8 =
Divide both sides by 4.
x= -2

10. Example 5
Solve.
3b – 2 = 2b + 12
3b – 2 = 2b + 12
– 2b – 2b
Subtract 2b from both sides.
b – 2 =
Add 2 to both sides.
b =

11. Example 6
Solve.
3w + 1 = 3w + 8
3w + 1 = 3w + 8
– 3w – 3w
Subtract 3w from both sides.
1 ≠
No solution. There is no number that can be substituted for the variable w to make the equation true.

12. 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.

13. Example 7: 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 z
Add 2z to both sides.
8z – 15 = – 7
Add 15 to both sides.
8z = 8
8z 8 8 =
Divide both sides by 8.
z = 1

14. Example 8
Solve.
12z – 12 – 4z = 6 – 2z + 32
12z – 12 – 4z = 6 – 2z + 32
8z – 12 = –2z + 38
Combine like terms.
+ 2z z
Add 2z to both sides.
10z – 12 =
Add 12 to both sides.
10z = 50
10z 10 =
Divide both sides by 10.
z = 5

15. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

16. Lesson Quiz
Solve.
1. 4x + 16 = 2x
2. 8x – 3 = x
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 1 4 1 2
x = 36
An orange has 45 calories. An apple has 75 calories.

17. Lesson Quiz for Student Response Systems
1. Combine like terms. 4p + 14 = 11p
A. p = 2
B. p = 7
C. p = 14
D. p = 15

18. Lesson Quiz for Student Response Systems
2. Combine like terms. 3g – 6 = 4g – 7
A. g = 1
B. g = 7
C. g = 13
D. g = –13

19. Lesson Quiz for Student Response Systems
3. Combine like terms. 3(g – 2) = 7g – 18
A. g = 2
B. g = 3
C. g = –3
D. g = –2