Introduction to Algebra
2.1 Introduction to Algebra – Cheat Sheet
What is Algebra?
- In algebra, letters (variables) are used to represent numbers.
- These letters can stand for unknown values, generalised numbers, or values that can change.
- Examples:
- x + 5 = 12 (x is unknown)
- A = l × w (A, l, w are generalised numbers in a formula)
Key Terms
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter representing a number | x, y, a |
| Coefficient | Number multiplying a variable | 5x (coefficient is 5) |
| Constant | Fixed value | 7, −3, 10 |
| Expression | Numbers, variables, and operations without an equals sign | 3x + 2y |
| Equation | Statement with an equals sign showing two expressions are equal | 2x − 3 = 7 |
| Formula | Mathematical rule linking variables | A = πr² |
Substituting into Expressions
- Replace the variable with the given number.
- Perform operations in the correct order (BODMAS).
| Example | Steps | Answer |
|---|---|---|
| 3x + 4, x = 2 | 3(2) + 4 | 6 + 4 = 10 |
| 2a² − b, a = 3, b = 5 | 2(3²) − 5 | 2(9) − 5 = 18 − 5 = 13 |
| 5xy, x = 2, y = 4 | 5(2)(4) | 40 |
| (m + n)², m = 1, n = 3 | (1 + 3)² | (4)² = 16 |
Substituting into Formulas
- Identify the formula.
- Replace each variable with the given number.
- Use the correct units if required.
| Formula | Given Values | Working | Answer |
|---|---|---|---|
| A = l × w | l = 5, w = 7 | 5 × 7 | 35 |
| C = 2πr | r = 3 | 2 × π × 3 | 6π |
| v = u + at | u = 10, a = 2, t = 4 | 10 + (2 × 4) | 18 |
| P = 2(l + w) | l = 8, w = 5 | 2(8 + 5) | 26 |
Tips for Substitution
- If the variable is squared or cubed, substitute first, then apply the power.
- Use brackets when substituting negative numbers.
- Example: If x = −2, x² = (−2)² = 4, not −4.
- For fractions, substitute carefully into numerator and denominator.