Equations with variables on both sides

The setup

Some equations have variables on both sides of the equals sign:

5x + 3 = 2x + 18

The strategy is to collect like terms: get all the variable terms on one side and all the numbers on the other.

Step by step

Example: 5x + 3 = 2x + 18

1. Subtract 2x from both sides: 3x + 3 = 18
2. Subtract 3: 3x = 15
3. Divide by 3: x = 5

Check: Left = 5(5) + 3 = 28; Right = 2(5) + 18 = 28 ✓

Tip: Move the smaller variable term. It keeps the coefficient positive.

With brackets

Example: 3(x − 4) = 2(x + 1)

1. Expand: 3x − 12 = 2x + 2
2. Subtract 2x: x − 12 = 2
3. Add 12: x = 14

With fractions

Multiply every term by the lowest common denominator.

Example: x/2 + 3 = x/3 + 5

1. × 6: 3x + 18 = 2x + 30
2. x = 12

With negatives

Example: 7 − 2x = 4x − 11

1. Add 2x: 7 = 6x − 11
2. Add 11: 18 = 6x
3. x = 3

Special cases

Example: 2(x + 3) = 2x + 6 → 2x + 6 = 2x + 6 → 6 = 6. Every x works.

Example: 4x + 1 = 4x − 5 → 1 = −5. No x works.

Word problem

Gym A charges $40/month plus a $60 fee. Gym B charges $50/month with no fee. When do they cost the same?

• 40m + 60 = 50m
• 60 = 10m
• m = 6 months

After 6 months, Gym A is cheaper.

Worked examples

Example 1: 8y − 5 = 3y + 20 → 5y = 25 → y = 5.

Example 2: 2(3a + 1) = 4a − 6 → 6a + 2 = 4a − 6 → 2a = −8 → a = −4.

Example 3: 0.5x + 7 = 1.5x − 3 → 10 = x → x = 10.

Common mistakes

• Moving a term without changing its sign. Use inverse operations on both sides.
• Expanding brackets incorrectly, especially with negatives.
• Stopping at 0 = 0 without saying "infinitely many solutions".

Try it yourself

1. 6x + 4 = 2x + 24
2. 9 − n = 2n − 6
3. 4(k − 2) = 2(k + 5)
4. 3x + 7 = 3x + 2
5. Plan A: $20 + $0.10/text. Plan B: $0.30/text. Same cost after how many texts?

Answers: 1) 5 2) 5 3) 9 4) no solution 5) 100

Quick recap

• Collect like terms: variables on one side, numbers on the other.
• Use inverse operations on both sides and check the answer.
• Watch for no solution and infinitely many solutions.