What is a linear equation in one variable?
An equation is a statement that two things are equal. It has an equals sign (=).
A variable (or unknown) is a letter, like x, that stands for a number we do not know yet.
An equation is linear when the variable has power 1. So 2x + 3 = 11 is linear. But x² = 9 is not linear.
The solution (or root) is the number that makes both sides equal. For 2x + 3 = 11, the solution is x = 4.
General form: ax + b = 0, where a ≠ 0. A linear equation in one variable has exactly one solution.
The balance method
Think of the equals sign as the middle of a balance. You may do any of these to both sides and it stays level:
- add the same number
- subtract the same number
- multiply by the same number
- divide by the same non-zero number
Goal: get x alone on one side.
Transposing (shortcut)
Moving a term to the other side and changing its sign is called transposing. It is just the balance method written faster.
x + 7 = 12 → x = 12 − 7 = 5. (+7 became −7.)
3x = 18 → x = 18 ÷ 3 = 6. (× 3 became ÷ 3.)
Harder equations
Variable on both sides
Collect all x terms on one side and all numbers on the other. 5x − 4 = 2x + 11 → 5x − 2x = 11 + 4 → 3x = 15 → x = 5.
Brackets
Open brackets first. 3(x − 2) = 12 → 3x − 6 = 12 → 3x = 18 → x = 6. Or divide both sides by 3 first: x − 2 = 4 → x = 6.
Fractions
Multiply both sides by the LCM of the denominators to clear fractions. x/2 + x/3 = 10 → multiply by 6 → 3x + 2x = 60 → 5x = 60 → x = 12.
Equations that reduce to linear form
(x + 1)/(x − 2) = 3/2. Cross-multiply: 2(x + 1) = 3(x − 2) → 2x + 2 = 3x − 6 → x = 8. (x cannot be 2, because we cannot divide by zero.)
Special cases
If x disappears and you get a true statement like 4 = 4, every number works (an identity). If you get a false one like 4 = 7, there is no solution.
Word problems: from story to equation
- Read carefully. What is unknown? Call it x (write it down: "Let x = …").
- Write the other amounts using x.
- Find the sentence that says two things are equal. Make the equation.
- Solve.
- Check the answer in the story, and write it with units.
Example: a rectangle is 4 m longer than it is wide. Its perimeter is 32 m. Let width = x. Length = x + 4. 2(x + x + 4) = 32 → 4x + 8 = 32 → x = 6. Width 6 m, length 10 m.
Common types: ages, consecutive numbers, perimeter, money and coins, speed–distance–time, percent.
Key formulas and definitions
- ax + b = 0 → x = −b/a (a ≠ 0)
- ax + b = c → x = (c − b)/a
- Transpose: + ↔ −, × ↔ ÷
- Clear fractions: multiply both sides by the LCM
- Check: put x back in both sides
Worked examples
1. Solve x − 9 = 4.
Add 9 to both sides: x = 4 + 9 = 13. Check: 13 − 9 = 4 ✓.
2. Solve 4x + 5 = 29.
Subtract 5: 4x = 24. Divide by 4: x = 6. Check: 4 × 6 + 5 = 29 ✓.
3. Solve 7x − 3 = 4x + 12.
Subtract 4x: 3x − 3 = 12. Add 3: 3x = 15. Divide by 3: x = 5. Check: 32 = 32 ✓.
4. Solve 2(3x − 1) = 5(x + 2).
Open brackets: 6x − 2 = 5x + 10. Subtract 5x: x − 2 = 10. So x = 12. Check: 2 × 35 = 70 and 5 × 14 = 70 ✓.
5. Solve x/4 − 1 = x/6.
LCM of 4 and 6 is 12. Multiply by 12: 3x − 12 = 2x. So x = 12. Check: 3 − 1 = 2 ✓.
6. The sum of three consecutive numbers is 72. Find them.
Let them be x, x + 1, x + 2. 3x + 3 = 72 → 3x = 69 → x = 23. Numbers: 23, 24, 25.
7. A mother is 3 times as old as her son. In 12 years she will be twice as old. Find their ages now.
Son = x, mother = 3x. In 12 years: 3x + 12 = 2(x + 12) → 3x + 12 = 2x + 24 → x = 12. Son 12 years, mother 36 years.
Common mistakes
- Doing a step to one side only. If you subtract 3 on the left, subtract 3 on the right too.
- Forgetting to change the sign when transposing: x + 5 = 9 gives x = 9 − 5, not 9 + 5.
- Multiplying only the first term inside a bracket: 3(x − 2) is 3x − 6, not 3x − 2.
- Clearing fractions but forgetting to multiply the whole-number terms too: in x/2 + 1 = 4, multiplying by 2 gives x + 2 = 8.