Equations, Factors & Formulae

📐 Algebra Essentials: Equations, Factors & Formulae

Welcome! This guide covers fundamental algebra concepts: solving equations (finding unknowns), factorization (breaking expressions into factors), and rearranging formulae (changing the subject). These skills apply to real-world problems from cooking times to physics calculations.

1️⃣ Solving Equations

Linear Equations

A linear equation has variables with power ≤ 1 (e.g., 3x + 1 = 13).

Golden Rule: Whatever you do to one side, do to the other side to keep it balanced.

Method 1: Function Machine

x → [×3] → 3x → [+1] → 3x+1 = 13
↓ Work backwards ↓
4 ← [÷3] ← 12 ← [-1] ← 13

Result: x = 4

Method 2: Algebraic Steps

Solve: 3x + 1 = 13

Step 1: 3x + 1 - 1 = 13 - 1 → 3x = 12
Step 2: 3x ÷ 3 = 12 ÷ 3 → x = 4

Type 1: Variable on Both Sides (Same Signs)

Solve: 5x - 2 = 3x + 6

Subtract 3x: 2x - 2 = 6
Add 2: 2x = 8
Divide by 2: x = 4

Type 2: Variable with Different Signs

Solve: 5x + 12 = 20 - 11x

Add 11x: 16x + 12 = 20
Subtract 12: 16x = 8
Divide by 16: x = 1/2

Type 3: Equations with Brackets

Solve: 2(y-4) + 4(y+2) = 30

Expand: 2y - 8 + 4y + 8 = 30
Simplify: 6y = 30
Result: y = 5

Remember BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction.

Type 4: Equations with Fractions

Solve: 6/p = 10 (where p is in denominator)

Multiply by 7: 6p = 70
Divide by 6: p = 35/3

Type 5: Exponential Equations

Solve: 3^(3x+1) = 81

Rewrite: 81 = 3^4
Equate powers: 3x + 1 = 4
Result: x = 1

Index Laws:
• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^mn
• a⁰ = 1

2️⃣ Factorising Algebraic Expressions

Factorization is the opposite of expanding brackets. We extract common factors to write expressions with brackets.

Finding the Highest Common Factor (HCF)

• HCF of 12 and 4 is 4
• HCF of 12x and 8x is 4x

Basic Method

Factorise: 12x - 4

HCF = 4
12x - 4 = 4(3x - 1)
✓ Check: 4(3x-1) = 12x - 4

Always check your answer by expanding the brackets!

Type 1: Numerical and Variable Factors

Factorise: 15x + 12y

HCF = 3
Result: 3(5x + 4y)

Factorise: 18mn - 30m

HCF = 6m
Result: 6m(3n - 5)

Type 2: Higher Powers

Factorise: 36pq² - 24p²q

HCF = 12pq
Result: 12pq(3q - 2p)

Type 3: Factorising by Grouping

Factorise: 6bx - 15cx + 10cy - 4by

Group: (6bx - 15cx) + (10cy - 4by)
Factor each: 3x(2b-5c) + 2y(5c-2b)
Rearrange: 3x(2b-5c) - 2y(2b-5c)
Result: (2b-5c)(3x-2y)

3️⃣ Rearranging Formulae (Changing the Subject)

The subject is the variable that stands alone on one side of the equation.

In v = u + at, the subject is v
In F = ma, the subject is F

Basic Rearrangement

Make u the subject: v = u + at

Subtract at: v - at = u
Rewrite: u = v - at

Type 1: Simple Operations

Make y the subject: x + y = c

Subtract x: y = c - x

Type 2: Dealing with Fractions

Make b the subject: a/b = d

Multiply by b: a = bd
Divide by d: b = a/d

Type 3: Square Roots and Powers

Make x the subject: √x + y = z

Subtract y: √x = z - y
Square both sides: x = (z - y)²

Real-World Application: Motion Formula

Original: v = u + at
Make t the subject:

v - u = at
t = (v - u)/a

Use: Calculate time when you know initial velocity (u), final velocity (v), and acceleration (a).

Application: Perimeter Formula

Original: P = 2(l + w)
Make w the subject:

P/2 = l + w
w = P/2 - l

🔑 Key Formulas Summary

BODMAS Order:
1. Brackets
2. Orders (powers/roots)
3. Division & Multiplication (left to right)
4. Addition & Subtraction (left to right)

Index Laws:
• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^mn
• a⁰ = 1

Pro Tips:
• Always line up '=' signs vertically for clarity
• Keep coefficients of x positive when possible
• Leave answers as fractions unless decimals are requested
• Always check factorization by expanding

📚 Master these fundamentals to excel in algebra!