📐 Mathematics: Algebra & Equations
ACER Level 1 · Expressions, equations, expanding, factorising & sequences
📋 What You’ll Learn in This Lesson
- Like terms, simplification and substitution with worked examples
- Solving linear equations step by step (including multi-step)
- Expanding brackets including FOIL method
- Factorising: common factors and difference of squares
- Arithmetic and geometric sequences – finding the rule
🎯 ACER Algebra principle: Show every step even in a multiple choice context. Writing out intermediate steps reduces sign errors by ~70% and helps you catch mistakes before committing to an answer.
📝 Like Terms & Simplification
| Expression |
Working |
Answer |
| 4x – 3y + 2x + 5y |
Collect x terms: 4x + 2x = 6x. Collect y terms: -3y + 5y = 2y. |
6x + 2y |
| 3a^2 + 2a – a^2 + 4a |
a^2 terms: 3a^2 – a^2 = 2a^2. a terms: 2a + 4a = 6a. |
2a^2 + 6a |
| (2x + 3) + (5x – 7) |
Collect: 2x + 5x = 7x. Constants: 3 – 7 = -4. |
7x – 4 |
🔧 Solving Linear Equations — Step by Step
📝 Worked Example — Multi-step equation
Question
Solve: 3(x – 4) = 15
Add 12 both sides
3x = 27
Check (always!)
3(9 – 4) = 3 x 5 = 15 ✓
✅ Answer: x = 9
📝 Worked Example — Equation with variable on both sides
Question
Solve: 5x + 3 = 2x + 12
Check
5(3) + 3 = 18 = 2(3) + 12 = 18 ✓
✅ Answer: x = 3
🔄 Expanding Brackets — FOIL Method
(a + b)(c + d) = ac + ad + bc + bd
First, Outside, Inside, Last
📝 Worked Example — FOIL expansion
Question
Expand: (x + 5)(x – 3)
Combine
x^2 – 3x + 5x – 15 = x^2 + 2x – 15
✅ Answer: x^2 + 2x – 15
⚠️ Watch Out: The #1 algebra error: sign mistakes in brackets. When multiplying a negative by a bracket: -2(x – 5) = -2x + 10 (NOT -2x – 10). The minus distributes to BOTH terms inside.
🔁 Factorising
| Type |
Method |
Example |
| Common factor |
Find the HCF of all terms |
6x^2 + 9x = 3x(2x + 3) |
| Difference of two squares |
a^2 – b^2 = (a-b)(a+b) |
x^2 – 9 = (x-3)(x+3) |
| Quadratic (if asked) |
Find two numbers that multiply to c and add to b in x^2+bx+c |
x^2 + 5x + 6 = (x+2)(x+3) |
🔢 Sequences — Finding the Rule
| Type |
How to identify |
Example + Rule |
| Arithmetic |
Common difference (add/subtract constant) |
3, 7, 11, 15… difference = +4. Rule: 4n – 1 (n starts at 1) |
| Geometric |
Common ratio (multiply/divide by constant) |
2, 6, 18, 54… ratio = x3. Rule: 2 x 3^(n-1) |
| Pattern test |
Subtract consecutive terms: constant? arithmetic. Divide? geometric. |
If differences of differences are constant: quadratic sequence |
⭐ ACER Examiner’s Trick: ACER sequence questions often ask for the nth term, not the next term. Memorise: Arithmetic nth term = a + (n-1)d, where a = first term and d = common difference.
📌 Quick Reference Card — Algebra & Equations
| Simplify |
Collect like terms. Variables must match exactly (x and x^2 are NOT like terms). |
| Solve equations |
Do the same operation to both sides. Always check by substituting back. |
| FOIL |
(a+b)(c+d) = ac + ad + bc + bd. Watch signs on last term. |
| Difference of squares |
x^2 – n^2 = (x-n)(x+n). Very common in ACER. |
| Arithmetic sequence rule |
nth term = first term + (n-1) x common difference |
| Sign rule |
-( x – y) = -x + y. NEVER -x – y. |
