GCSE Maths Simultaneous & inequalities revision

For 2x + y = 11 and 3x − y = 9, adding the equations cancels y: 5x = 20, so x = 4 and (from 2 × 4 + y = 11) y = 3. When the coefficients do not match, scale one equation first: 4x + 3y = 10 and 2x + 3y = 8 subtract to give 2x = 2, so x = 1 and y = 2.

Sometimes both equations need scaling. For 3x + 2y = 12 with 5x − 2y = 4, adding works directly (8x = 16, x = 2, y = 3). But for 2x + 3y = 12 and 3x + 4y = 17, multiply the first by 3 and the second by 2: 6x + 9y = 36 and 6x + 8y = 34, so y = 2 and x = 3. Always finish by checking in both original equations — 2(3) + 3(2) = 12 ✓ and 3(3) + 4(2) = 17 ✓.

When an equation reads y = x + 1, substitute into the other: 3x + 2y = 12 becomes 3x + 2(x + 1) = 12, so 5x = 10, x = 2, y = 3. Substitution is also the method for a linear–quadratic pair (Higher): with y = 2x and x² + y² = 45, substitute to get x² + 4x² = 45, so x² = 9 and x = 3, y = 6 or x = −3, y = −6 — two crossing points, both valid.

Another Higher staple: y = x² and y = 3x + 4? No — take y = x² with y = x + 2: x² = x + 2 gives x² − x − 2 = 0, so (x − 2)(x + 1) = 0 and the solution pairs are (2, 4) and (−1, 1). Each x needs its own y — quoting one y for both x’s loses marks.

Inequalities solve exactly like equations except when you multiply or divide by a negative: −2x ≥ 6 means x ≤ −3, sign flipped. For 3x + 2 < 11, subtract and divide to get x < 3. Compound inequalities squeeze from both sides: 5 ≤ 2x + 1 < 9 means 4 ≤ 2x < 8, so 2 ≤ x < 4.

On a number line, strict inequalities get an open circle (x < 3) and inclusive ones a filled circle (x ≥ 2). Integer lists come from the picture: the integers satisfying 2 ≤ x < 4 are just 2 and 3. In words: “x is at least 2 but less than 4” — the ends are described by the symbols, not by guesswork.

Higher. A quadratic inequality splits the line at its roots. x² − 5x + 6 ≤ 0: roots 2 and 3, and the quadratic is negative between them, so 2 ≤ x ≤ 3. x² − 4 > 0: roots ±2, and the quadratic is positive outside, so x < −2 or x > 2. Sketching the parabola turns every such question into a picture: below the axis between the roots, above it outside.

Regions in two variables: y < x + 2 is everything below the dashed line y = x + 2 (dashed because equality is excluded); y ≥ 2x − 1 is the solid line plus the region above it. For a pair of inequalities, shade both and keep the overlap. Set notation writes the answers: {x : x > 3}, or {x : 2 ≤ x < 4} for a compound — the colon reads “such that”.