Algebra

What is Algebra?

Algebra is the branch of mathematics that uses letters (variables) to represent unknown or changing values. Instead of working only with specific numbers, algebra allows you to write general rules that work for any number.

For example, the rule "double a number and add 3" can be written as 2n + 3, where n stands for any number you choose. This compact notation is one of the most powerful tools in all of mathematics.

Algebra appears in every part of higher maths — from geometry formulas to statistics and calculus — so mastering the basics here pays dividends throughout your studies.

Key Terms

Term	Meaning	Example
Variable	A letter representing an unknown value	x, y, n
Constant	A fixed number that does not change	5, -3, 100
Term	A single number, variable, or their product	4x, -2, 7y²
Expression	A combination of terms without an equals sign	3x + 2
Equation	A statement that two expressions are equal	3x + 2 = 11
Coefficient	The number multiplying a variable	4 in 4x
Like terms	Terms with the same variable and power	3x and 5x

Collecting Like Terms

Like terms share the same variable raised to the same power. You can add or subtract them to simplify an expression.

Example: Simplify 5x + 3y - 2x + 4y

• Collect x terms: 5x - 2x = 3x
• Collect y terms: 3y + 4y = 7y
• Answer: 3x + 7y

You cannot combine unlike terms (e.g. 3x and 4y) — they stay separate.

Solving One-Step Equations

To solve an equation, find the value of the variable that makes it true. Use inverse operations — the opposite of whatever is being done to the variable.

Example: x + 7 = 15

• Subtract 7 from both sides: x = 15 - 7 = 8

Example: 3x = 18

• Divide both sides by 3: x = 18 ÷ 3 = 6

The golden rule: whatever you do to one side, do to the other.

Solving Two-Step Equations

Two-step equations require two inverse operations. Work backwards through the order of operations.

Example: 2x + 4 = 12

• Step 1 — subtract 4 from both sides: 2x = 8
• Step 2 — divide both sides by 2: x = 4

Example: 3x - 5 = 16

• Step 1 — add 5 to both sides: 3x = 21
• Step 2 — divide both sides by 3: x = 7

Always perform the addition/subtraction step before the multiplication/division step when working backwards.

The Distributive Property

The distributive property allows you to expand brackets by multiplying the term outside by each term inside.

Rule: a(b + c) = ab + ac

Example: 4(2x - 3) = 4 × 2x - 4 × 3 = 8x - 12

Example: -3(x + 5) = -3x - 15

Watch the signs carefully when the factor outside the bracket is negative.

Substitution

Substitution means replacing a variable with a given number and calculating the result.

Example: Find the value of 3x² - 2x + 1 when x = 3

• 3(3²) - 2(3) + 1
• 3(9) - 6 + 1
• 27 - 6 + 1 = 22

Substitution is used constantly — in checking solutions, evaluating formulas, and function notation.

Order of Operations (BODMAS)

When an expression contains mixed operations, follow BODMAS:

Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction

Example: 3 + 4 × 2 = 3 + 8 = 11 (multiply before adding)

Example: (3 + 4) × 2 = 7 × 2 = 14 (brackets first)

A common mistake is reading left to right and ignoring the hierarchy — BODMAS prevents this.

Forming Equations from Word Problems

Algebra becomes truly useful when you translate real-world problems into equations.

Example: "I think of a number, multiply it by 5 and subtract 3. The result is 22. What is the number?"

• Let the number be n
• Form the equation: 5n - 3 = 22
• Solve: 5n = 25, so n = 5

Key phrases: "a number" → variable; "is/equals" → = sign; "more than" → +; "less than" → -.

Common Mistakes

• Adding unlike terms — 3x + 4y cannot be simplified to 7xy; they are different terms.
• Forgetting to apply operations to both sides — if you add 5 to the left, you must add 5 to the right.
• Sign errors when expanding — be especially careful with negative factors: -2(x - 3) = -2x + 6, not -2x - 6.
• Ignoring BODMAS — always evaluate brackets and powers before multiplication, division, addition, or subtraction.
• Substituting incorrectly — when x = -2, x² = (-2)² = 4, not -4.

Tips and Tricks

• Check your solution by substituting it back into the original equation and verifying both sides are equal.
• Write every step — even experienced mathematicians write out each line to avoid errors.
• Draw a balance scale in your mind: equations stay balanced only when you do the same thing to both sides.
• For multi-step problems, label what the variable represents at the start (e.g. "Let x = the unknown number") to keep your working clear.