Inequalities
2.5 Equations – Cheat Sheet
1. Constructing Expressions, Equations, and Formulas
- Expression: Combination of variables, numbers, and operations (no equals sign).
- Example: 2x + 3y
- Equation: States two expressions are equal.
- Example: 3x + 4 = 10
- Formula: Rule linking variables.
- Example: A = l × w
- Example Question: Product of two consecutive even numbers = n × (n + 2)
2. Solving Linear Equations in One Unknown
- Isolate the variable using inverse operations.
| Example | Working | Answer |
|---|---|---|
| 3x + 4 = 10 | 3x = 6 → x = 2 | x = 2 |
| 5 − 2x = 3(x + 7) | 5 − 2x = 3x + 21 → −2x − 3x = 21 − 5 → −5x = 16 → x = −16/5 | x = −3.2 |
3. Solving Fractional Equations
Step-by-Step Method:
- Find the Lowest Common Denominator (LCD).
- Multiply through by the LCD to remove fractions.
- Solve as a normal equation.
| Example | Working | Answer |
|---|---|---|
| x / (2x + 1) = 4 | Multiply by (2x + 1): x = 4(2x + 1) → x = 8x + 4 → −7x = 4 → x = −4/7 | x = −4/7 |
| 2/(x + 2) + 3/(2x − 1) = 1 | LCD = (x + 2)(2x − 1) → Multiply through, simplify, solve quadratic | Depends on quadratic roots |
4. Solving Simultaneous Linear Equations
Substitution Method:
- Solve one equation for one variable, substitute into the other.
Elimination Method:
- Multiply equations if needed, add/subtract to eliminate a variable.
| Example | Working | Answer |
|---|---|---|
| x + y = 5, 2x − y = 1 | Add: 3x = 6 → x = 2 → y = 3 | (2, 3) |
5. Solving Quadratic Equations
Factorisation:
- x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = −2 or x = −3
Completing the Square:
- x² + 6x + 5 = 0 → (x + 3)² − 9 + 5 = 0 → (x + 3)² = 4 → x = −3 ± 2
Quadratic Formula:
x = [−b ± √(b² − 4ac)] / (2a)
| Example | a | b | c | Discriminant | Roots |
|---|---|---|---|---|---|
| x² + 6x + 5 = 0 | 1 | 6 | 5 | 36 − 20 = 16 | −3 ± 2 → x = −1, −5 |
6. Changing the Subject of a Formula
Example 1: Subject appears once
- v = u + at → a = (v − u) / t
Example 2: Subject appears twice
- P = x + yx → P − yx = x → P = x(1 + y) → x = P / (1 + y)
Example 3: Subject with powers or roots
- A = πr² → r² = A / π → r = √(A / π)
7. Examples Table for Quick Reference
| Problem | Method | Answer |
|---|---|---|
| 3x + 4 = 10 | Linear | x = 2 |
| 5 − 2x = 3(x + 7) | Linear | x = −16/5 |
| x/(2x + 1) = 4 | Fractional | x = −4/7 |
| x + y = 5, 2x − y = 1 | Simultaneous | (2, 3) |
| x² + 5x + 6 = 0 | Factorisation | x = −2, −3 |
| x² + 6x + 5 = 0 | Completing square | x = −1, −5 |
| v = u + at | Change subject | a = (v − u)/t |