Solving a linear equation means finding the value of the letter that makes both sides equal. The method is to do the same thing to both sides until the letter stands alone.
How does the balance method work?
Think of the equals sign as a balanced scale. Whatever you add, subtract, multiply or divide on one side, you do to the other, so the scale stays level.
This explains the sign change. Moving +4 “across” really means subtracting 4 from both sides.
Worked example
Solve 5x + 4 = 3x + 16. This worked example has letters on both sides.
- Subtract 3x from both sides: 5x − 3x + 4 = 16, so 2x + 4 = 16.
- Subtract 4 from both sides: 2x = 12.
- Divide both sides by 2: x = 6.
Substitute to check. The left side is 5 × 6 + 4 = 34. The right side is 3 × 6 + 16 = 34. Both equal 34, so x = 6 is correct.
Now an example with brackets: 2(x − 3) = 10. Expand to get 2x − 6 = 10.
Add 6 to both sides: 2x = 16. Divide by 2: x = 8. Check: 2(8 - 3) = 10. Correct.
What does the common mistake look like?
A student solves 3x − 7 = 11 and writes 3x = 11 − 7 = 4, so x = 4 ÷ 3. The error is that −7 moved across without changing sign. To cancel −7 you add 7 to both sides.
The correct working is 3x = 11 + 7 = 18, so x = 6. Check: 3 × 6 − 7 = 11. The wrong answer fails the check instantly, because 3 × (4 ÷ 3) − 7 = −3, not 11.
Steps to follow
- Expand any brackets, then collect like terms on each side.
- Add or subtract to get the letter terms on one side and the plain numbers on the other.
- Divide by the number in front of the letter, then substitute to check.
Self-check
Solve 7 − 2x = 3x − 8.
Answer
Add 2x to both sides: 7 = 5x − 8. Add 8 to both sides: 15 = 5x. Divide by 5: x = 3.
Check: the left side is 7 − 2 × 3 = 1, and the right side is 3 × 3 − 8 = 1. Both equal 1, so x = 3 is correct. A common slip is to keep the letter negative, which leads to x = −3.
Where to go next
The same balance idea applies to formulae, which is the focus of rearranging a formula with fractions. If brackets are the trouble spot, go back to expanding and factorising simple expressions. The algebra readiness hub lists every skill, and Mathematics tuition is there if you want a teacher to watch your working.