Algebra

Understanding Algebra

Algebra is the branch of mathematics where we use letters (called variables) to represent unknown numbers. Think of it as a powerful tool that lets us solve problems even when we don't know all the values yet.

In algebra, letters like x, y, and n stand in for numbers we're trying to find or numbers that can change. For example, if you buy x apples at 30p each, the total cost would be 30x pence. This simple idea forms the foundation of all algebraic thinking.

Algebra matters because it helps us describe patterns, solve real-world problems, and work with relationships between quantities. Whether you're calculating phone bills, working out journey times, or predicting savings, algebra gives you the tools to find answers systematically.

Key skills in algebra include: simplifying expressions (combining like terms), expanding brackets, factorising, solving equations, and working with formulas. You'll also need to substitute numbers into expressions and rearrange formulas to find different variables.

Remember: algebra follows the same rules as arithmetic. The letters behave exactly like numbers would. If you can add 3 + 3 to get 6, then you can add 3x + 3x to get 6x. Building confidence with these basics will help you tackle more complex algebraic problems.

Key formulas

• 3x + 5x = 8x — Combine terms with the same letter and power
• a(b + c) = ab + ac — Multiply everything inside the bracket by the term outside
• ab + ac = a(b + c) — Take out the common factor and put remaining terms in brackets
• If ax + b = c, then x = (c - b) ÷ a — Isolate the variable by doing inverse operations

Worked example

Question: Simplify 4x + 3y - 2x + 5y, then solve 3x + 7 = 22

Part 1: Simplifying 4x + 3y - 2x + 5y

Step 1: Group the like terms together

x terms: 4x - 2x = 2x

y terms: 3y + 5y = 8y

Step 2: Write the simplified expression

Answer: 2x + 8y

Part 2: Solving 3x + 7 = 22

Step 1: We need to get x on its own. First, remove the +7 by subtracting 7 from both sides

3x + 7 - 7 = 22 - 7

3x = 15

Step 2: Now divide both sides by 3 to find x

3x ÷ 3 = 15 ÷ 3

x = 5

Step 3: Check your answer by substituting back

3(5) + 7 = 15 + 7 = 22 ✓

Final Answer: x = 5

Common mistakes to avoid

• Forgetting that 5x means 5 × x, not 5 + x. So 5x when x = 3 equals 15, not 8.

• Combining unlike terms: you cannot simplify 3x + 4y into 7xy. Different letters (or powers) must stay separate.

• Sign errors when subtracting: remember that -2x - 3x = -5x, not -x or +5x. Watch your negative signs carefully!

• Not doing the same operation to both sides of an equation. Whatever you do to the left side, you must do to the right side to keep the equation balanced.

Exam tips

• Always show your working step by step. Even if you make a calculation error, you can still earn method marks for correct algebraic steps.

• When simplifying, write out the like terms you're combining. This helps you avoid errors and shows the examiner your method clearly.

• Check your solutions by substituting your answer back into the original equation. If both sides are equal, you've got it right!

• Read the question carefully: 'simplify' means collect like terms, 'expand' means multiply out brackets, 'factorise' means put into brackets, and 'solve' means find the value of the unknown.