Equations as balance scales
An equation says two sides are equal, like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced.
To solve an equation, you isolate the variable: get it by itself on one side.
The key idea: inverse operations
Undo what’s being done to the variable using the inverse operation.
| Equation | Variable is… | Undo by… | Solution |
|---|---|---|---|
| x + 7 = 12 | added to 7 | subtracting 7 | x = 5 |
| x − 4 = 9 | having 4 taken away | adding 4 | x = 13 |
| 3x = 18 | multiplied by 3 | dividing by 3 | x = 6 |
| x/5 = 4 | divided by 5 | multiplying by 5 | x = 20 |
Showing your steps
Example: x + 7 = 12
x + 7 − 7 = 12 − 7
x = 5
Example: 3x = 18
3x ÷ 3 = 18 ÷ 3
x = 6
Check your solution
Substitute the solution back into the original equation.
- x = 5: 5 + 7 = 12 ✓
Negatives and fractions
- −2x = 10 → divide by −2 → x = −5
- x + 9 = 4 → subtract 9 → x = −5
- (2/3)x = 8 → multiply by 3/2 → x = 12
- 0.5x = 3.5 → divide by 0.5 → x = 7
Variable on the right
The variable can be on either side: 15 = y − 6 → add 6 → 21 = y, so y = 21.
Writing equations from words
Problem: A number increased by 14 is 30. Find the number.
- n + 14 = 30
- n = 16
Problem: Five tickets cost $45. How much is one?
- 5t = 45
- t = $9
Worked examples
Example 1: m − 8 = −3
Add 8: m = 5.
Example 2: k/4 = 7
Multiply by 4: k = 28.
Example 3: −6 = 3p
Divide by 3: p = −2.
Common mistakes
- Using the same operation instead of the inverse: for x + 7 = 12, adding 7.
- Only changing one side.
- Sign errors with negative coefficients.
- Not checking the answer.
Try it yourself
- x + 11 = 20
- y − 6 = −2
- 7n = 63
- a/3 = −5
- 2.5x = 10
Answers: 1) 9 2) 4 3) 9 4) −15 5) 4
Quick recap
- Isolate the variable using the inverse operation.
- Do the same thing to both sides to keep balance.
- Check the solution by substituting.