1. Warm Up Solve. 1. 2x + 9x – 3x + 8 = 16 2. –4 = 6x + 22 – 4x 3. + = 5
= 5
– = 3
x = 1
x = –13 2 7 x 7 7 1
x = 34 9x 16 2x 4 1 8
x = 50

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

3. 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.

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

5. A literal equation is an equation with two or more variables
A literal equation is an equation with two or more variables. A formula is one type of literal equation.

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

7. 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
Add 6 to both sides.
4b = 24 4b 4 24 =
Divide both sides by 4.
b = 6

8. Additional Example 1C: 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.

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

10. Check It Out: Example 1A
Solve.
5x + 8 = x
5x + 8 = x
– 5x – 5x
Subtract 5x from both sides.
8 = –4x 8 –4
–4x =
Divide both sides by –4.
–2 = x

11. Check It Out: Example 1C
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. 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 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