1. Solve Equations with Variables on Both Sides
Chapter 2.5
Solve Equations with Variables on Both Sides

2. Key Concept
Some equations have variables on both sides. To solve such equations, you can collect the variable terms on one side of the equation and the constant terms on the other side of the equation.

3. Solve an equation with variables on both sides
EXAMPLE 1
Solve an equation with variables on both sides
Solve 7 – 8x = 4x – 17.
7 – 8x = 4x – 17
Write original equation.
7 – 8x + 8x = 4x – x
Add 8x to each side.
7 = 12x – 17
Simplify each side.
24 = 12x
Add 17 to each side.
2 = x
Divide each side by 12.
ANSWER
The solution is 2. Check by substituting 2 for x in the original equation.

4. Solve an equation with variables on both sides
EXAMPLE 1
Solve an equation with variables on both sides
CHECK
7 – 8x = 4x – 17
Write original equation.
7 – 8(2) = 4(2) – 17 ?
Substitute 2 for x.
–9 = 4(2) – 17 ?
Simplify left side.
–9 = –9
Simplify right side. Solution checks.

5. Solve an equation with grouping symbols
EXAMPLE 2
Solve an equation with grouping symbols
9x – 5 = 1 4
(16x + 60).
Solve 1 4
(16x + 60)
9x – 5 =
Write original equation.
9x – 5 = 4x + 15
Distributive property
5x – 5 = 15
Subtract 4x from each side.
5x = 20
Add 5 to each side.
x = 4
Divide each side by 5.

6. On Your Own GUIDED PRACTICE Solve the equation. Check your solution.
– 3m = 5m 3
ANSWER

7. On Your Own GUIDED PRACTICE Solve the equation. Check your solution.
c = 4c – 7
ANSWER 9

8. On Your Own GUIDED PRACTICE Solve the equation. Check your solution.
– 3k = 17k – 2k
ANSWER –8

9. On Your Own GUIDED PRACTICE Solve the equation. Check your solution.
z – 2 = 2(3z – 4)
ANSWER 6

10. On Your Own GUIDED PRACTICE Solve the equation. Check your solution.
– 4a = 5(a – 3)
ANSWER 2

11. On Your Own GUIDED PRACTICE Solve the equation. Check your solution.2 3
(6y + 15) 6.
ANSWER 4

12. NUMBER OF SOLUTIONS
Equations do not always have one solution. An equation that is true for all values of the variable is an identity . So, the solution of an identity is all real numbers. Some equations have no solution.

13. Identify the number of solutions of an equation
EXAMPLE 4
Identify the number of solutions of an equation
Solve the equation, if possible.
a. 3x = 3(x + 4)
b. 2x + 10 = 2(x + 5)
SOLUTION
a. 3x = 3(x + 4)
Original equation
3x = 3x + 12
Distributive property
The equation 3x = 3x + 12 is not true because the number 3x cannot be equal to 12 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation.

14. Identify the number of solutions of an equation
EXAMPLE 4
3x – 3x = 3x + 12 – 3x
Subtract 3x from each side.
0 = 12
Simplify.
ANSWER
The statement 0 = 12 is not true, so the equation has
no solution.

15. Identify the number of solutions of an equation
EXAMPLE 1
EXAMPLE 4
Identify the number of solutions of an equation
b. 2x + 10 = 2(x + 5)
Original equation
2x + 10 = 2x + 10
Distributive property
ANSWER
Notice that the statement 2x + 10 = 2x + 10 is true for all values of x. So, the equation is an identity, and the solution is all real numbers.

16. On Your Own GUIDED PRACTICE Solve the equation, if possible.
z + 12 = 9(z + 3)
ANSWER
no solution

17. On Your Own GUIDED PRACTICE Solve the equation, if possible.
w + 1 = 8w + 1
ANSWER

18. On Your Own GUIDED PRACTICE Solve the equation, if possible.
(2a + 2) = 2(3a + 3)
ANSWER
identity