Year 12: Algebra

Year 12 algebra sits underneath almost every other topic. Clean manipulation of expressions, confident work with indices, and careful solving of simultaneous systems all feed directly into functions, calculus and probability.

Start by simplifying the structure of the problem. Factor, cancel and list restrictions before you do any heavier algebra.

What to master

Simplifying rational and index expressions without losing domain restrictions.
Using polynomial division and factorisation to rewrite higher-degree expressions.
Solving simultaneous systems, including cases where one equation is nonlinear.
Recognising arithmetic sequences and using the partial sum formula efficiently.

Exam habit: when you solve simultaneous nonlinear equations, substitute into one clean equation first, then check every candidate solution in the original pair. Extraneous answers are common.

Algebra tutorials

How to think about Year 12 algebra

Algebra at this level is not really about doing longer manipulation for its own sake. The real goal is to recognise structure. A rational expression asks you to think about factors and restrictions. A simultaneous nonlinear system asks you to think about substitution and the shape of the equations you are combining. A series question asks you to identify the pattern before reaching for a formula. If you train yourself to classify the structure first, the algebra becomes far less messy. This is also why good algebra supports so many other topics. Functions, calculus, exponentials, and even probability often become hard only because the algebra inside them has not been organised well.

Worked example

Problem: Solve the simultaneous system

y = x + 1

and

x^{2} + y = 7

Substitute $y = x + 1$ into the second equation: $x^{2} + x + 1 = 7$ .
Rearrange: $x^{2} + x - 6 = 0$ .
Factor: $(x + 3) (x - 2) = 0$ .
So $x = - 3$ or $x = 2$ .
Substitute back into $y = x + 1$ .
If $x = - 3$ , then $y = - 2$ .
If $x = 2$ , then $y = 3$ .
Check both pairs in the original nonlinear equation. Both work.

Answer: the solutions are

(- 3, - 2)

and

(2, 3)

Common traps

Cancelling factors before stating the values that would make the denominator zero.
Using a valid factorisation but then forgetting to check every final answer in the original question.
Applying a formula too early when a simpler algebraic rearrangement would have exposed the structure.
Losing signs when working with rational indices or negative powers.

Study routine

A reliable algebra revision habit is to annotate each line with its purpose. Are you factoring, making a common denominator, isolating a variable, or checking restrictions? That makes your working easier to audit and helps you spot where errors begin. In Year 12, accuracy usually comes from organisation more than from speed.

The best final check in algebra is to ask whether every line still means the same thing as the one before it. If you divided by an expression, did you note when it could be zero? If you squared both sides, did you remember that extra solutions may appear? That habit of monitoring equivalence is what turns algebra from guesswork into controlled reasoning.

Revision focus

A strong Year 12 habit is to move between representations instead of trusting just one. In algebra, that means checking whether each step preserves the same meaning. In functions and trigonometry, it means linking equations, graphs, intervals, and diagrams. In coordinate geometry, it means asking whether the computed coordinates still match the geometric story in the question. When several representations agree, your answer is usually on solid ground.

It also helps to add one final sentence of interpretation after the working is done. State what the solution means in the language of the problem, not just in symbols. That last step slows you down just enough to catch sign errors, missing restrictions, interval mistakes, or values that are mathematically correct but contextually impossible.