Solve two simultaneous linear equations by elimination and substitution; solve a linear and a quadratic equation simultaneously (Higher tier); and interpret the solution as the point of intersection.

What this dot point is asking

OCR reference A8 asks you to solve two simultaneous linear equations by elimination and substitution, and at Higher tier to solve one linear and one quadratic equation together. "Simultaneous" means both equations hold at once, so the solution is the pair of values satisfying both, which graphically is the point (or points) where the lines or curves intersect. The topic is tested on every tier and is a reliable source of multi-mark questions, including on the non-calculator paper.

Solving by elimination

Elimination removes one unknown by combining the equations.

For $2x + 3y = 13$ and $2x + y = 7$, the $x$ coefficients already match, so subtract: $(2x + 3y) - (2x + y) = 13 - 7$ gives $2y = 6$, so $y = 3$, then $2x + 3 = 7$ gives $x = 2$. When neither pair matches, scale first: to solve $3x + 2y = 12$ and $2x + 5y = 8$, multiply the first by $2$ and the second by $3$ so the $x$ terms both become $6x$. The equations become $6x + 4y = 24$ and $6x + 15y = 24$; subtracting gives $11y = 0$, so $y = 0$ and then $x = 4$.

The decision to add or subtract depends on the signs of the matched terms. If they are identical (both $+6x$), subtracting eliminates them. If they are opposite (one $+2y$, one $-2y$), adding eliminates them. A reliable habit is to label the equations, show the scaling, and write "subtract" or "add" explicitly, because OCR awards a method mark for a correct elimination strategy even before the arithmetic.

Solving by substitution

Substitution puts one equation inside the other.

Rearrange one equation to make a single unknown the subject, then substitute that expression into the other equation. For $y = 2x - 1$ and $3x + y = 14$, substitute to get $3x + (2x - 1) = 14$, so $5x - 1 = 14$, $x = 3$, and $y = 2(3) - 1 = 5$. Substitution is the natural choice when one equation is already in the form $y = \ldots$ or $x = \ldots$, because no rearranging is needed.

When you substitute a bracketed expression, keep it in brackets so that the multiplication or sign applies to the whole thing. For example, substituting $y = 3 - x$ into $2x - 3y = 5$ gives $2x - 3(3 - x) = 5$, which expands to $2x - 9 + 3x = 5$, so $5x = 14$ and $x = 2.8$. Dropping the bracket and writing $2x - 3 \times 3 - x$ would lose the sign on the second term, a frequent slip that the bracket prevents.

Linear and quadratic together (Higher)

When one equation is quadratic, substitution is the only route.

The graphical meaning

Each linear equation is a straight line; the solution pair is the point where the two lines cross. Two parallel lines never meet, which corresponds to no solution. A linear and a quadratic can cross twice (two solutions), touch once (one repeated solution) or miss entirely (no real solutions), mirroring the discriminant of the quadratic that results. OCR rewards stating both coordinates of every intersection point, so always pair each $x$ with its $y$.