WJEC GCSE Mathematics Algebra: a complete overview of manipulation, equations, sequences, graphs, inequalities and quadratics

What the Algebra content demands

Algebra is the language the rest of the course is written in. WJEC uses it for formulae in geometry, for rates in proportion and for modelling real problems, so weak algebra leaks marks everywhere. The content runs from manipulating expressions through solving equations, sequences, graphs and inequalities to simultaneous equations and, at Higher tier, quadratics. Because WJEC's Unit 1 is the non-calculator paper, confident written manipulation is essential, and the long components reward setting out clear, line-by-line working that secures method marks.

This guide walks through the Algebra content and ties together the matching dot-point pages, each of which has its own practice questions.

Algebraic manipulation

Collect like terms by adding coefficients of matching letters. Expand a single bracket by multiplying every inside term; expand double brackets with FOIL. Factorise by taking out the highest common factor, the reverse of expanding. Substitute by replacing letters with values and applying BIDMAS, using brackets around negatives. Change the subject of a formula by applying inverse operations to both sides until the wanted letter stands alone, undoing powers and roots last.

Solving linear equations

Solve by inverse operations, keeping the equation balanced. Expand brackets and clear fractions (multiply through by the denominator) first. When the unknown is on both sides, gather the unknowns on one side and the numbers on the other, moving the smaller unknown term to keep a positive coefficient. Form equations from worded problems by naming the unknown and translating each statement, then solve and check by substitution.

Sequences

A linear sequence changes by a constant common difference $d$, with nth term $dn + c$ where $c$ is the first term minus $d$. The nth term lets you find any term directly and test whether a number is in the sequence by solving for $n$ and checking it is a positive whole number. Recognise special sequences too: square, cube, triangular, geometric and Fibonacci-type, and at Higher the nth term of a simple quadratic sequence.

Straight line graphs

A straight line is $y = mx + c$, with gradient $m$ and y-intercept $c$. Gradient is the change in $y$ over the change in $x$. Plot from a table of values, or use the intercept and gradient directly. Rearrange any equation into $y = mx + c$ before reading off $m$ and $c$. At Higher, parallel lines share a gradient and perpendicular gradients multiply to $-1$ (negative reciprocals).

Inequalities

Solve like equations but keep the inequality sign, and reverse the sign whenever you multiply or divide by a negative. Show solutions on a number line with an open circle for strict inequalities and a closed circle for inclusive ones, plus an arrow for the direction. Solve double inequalities by operating on all three parts, and list integer solutions carefully, watching whether each boundary is included or excluded.

Simultaneous equations

Solve a pair of linear equations together. Elimination scales the equations so one variable matches, then adds (opposite signs) or subtracts (same sign) to remove it. Substitution rearranges one equation and substitutes into the other. The graphical solution is the point where the two lines cross. Form simultaneous equations from worded contexts by naming two unknowns and writing two relationships, then solve and check both values.

Quadratic equations and graphs (Higher)

A quadratic $ax^2 + bx + c = 0$ solves by factorising (two numbers multiplying to $c$ and adding to $b$), by the formula $x = \tfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, or by completing the square to $(x + p)^2 + q$. The graph is a parabola crossing the x-axis at the roots, the y-axis at $c$, with a turning point on its line of symmetry. The discriminant $b^2 - 4ac$ counts the roots.

Check your knowledge

A mix of manipulation, equation, sequence, graph and quadratic questions covering the Algebra content. Attempt them under timed conditions, then check against the solutions.

1. Expand and simplify $2(3x - 1) + 4(x + 2)$. (2 marks)
2. Solve $4x - 7 = 2x + 5$. (3 marks)
3. Factorise $6x^2 + 9x$. (1 mark)
4. Find the nth term of $4, 7, 10, 13, \ldots$ (2 marks)
5. Find the gradient and y-intercept of $y = 5 - 3x$. (2 marks)
6. Solve $3x + 1 \le 16$. (2 marks)
7. Solve $x^2 - x - 12 = 0$ by factorising. (3 marks)
8. Make $t$ the subject of $v = u + at$. (2 marks)