Linear Equations – One Variable, One Answer
A linear equation contains a variable raised to the power of 1 and produces a straight-line graph. Solving one means isolating the variable on one side of the equals sign by performing the same operations on both sides.
What Makes an Equation Linear?
A linear equation has no powers higher than 1 on the variable. Every linear equation in one variable has exactly one solution. Examples: 3x + 5 = 11 and 7 - 2y = 1.
Solving One-Step Equations
| Equation | Operation Needed | Solution |
|---|---|---|
| x + 8 = 13 | Subtract 8 from both sides | x = 5 |
| x - 4 = 9 | Add 4 to both sides | x = 13 |
| 5x = 35 | Divide both sides by 5 | x = 7 |
| x / 3 = 6 | Multiply both sides by 3 | x = 18 |
Solving Two-Step Equations
Step 1 — subtract 3: 2x = 8. Step 2 — divide by 2: x = 4. Check: 2(4) + 3 = 11. Correct.
Add 7: 5y = 25. Divide by 5: y = 5. Check: 5(5) - 7 = 18. Correct.
Variables on Both Sides
Subtract x: 3x + 3 = 12. Subtract 3: 3x = 9. Divide by 3: x = 3. Check: 4(3)+3=15 and 3+12=15. Correct.
Add 2m: 7 = 5m - 3. Add 3: 10 = 5m. Divide by 5: m = 2.
Equations with Fractions
Subtract 1: x/4 = 4. Multiply by 4: x = 16.
Multiply by 3: 2x + 1 = 15. Subtract 1: 2x = 14. Divide by 2: x = 7.
Real-Life Application
A taxi costs a fixed fare of 3 plus 2 per kilometre. The total cost is 17. How far was the journey?
3 + 2d = 17. 2d = 14. d = 7 km.
Common Mistakes
- Only doing an operation to one side of the equation.
- Moving a term across the equals sign without changing its sign.
- Dividing by a coefficient but forgetting to divide the entire side.
Key Takeaways
- Isolate the variable by performing inverse operations equally on both sides.
- Work backwards through the order of operations.
- Always verify your answer by substituting back into the original equation.
Practice Questions
- Solve 3x - 4 = 11.
- Solve 8 + 2y = 22.
- Solve 5a - 3 = 2a + 9.
- Solve (3p - 2) / 4 = 2.
- A number is doubled and then 5 is subtracted, giving 19. Write and solve the equation to find the number.