The substitution method
When one equation is linear and the other is non-linear (containing x², y² or xy), we use substitution. Make one variable the subject in the linear equation, substitute it into the non-linear one, and simplify to a single quadratic equation. Solving that quadratic gives the x-values; back-substitute into the linear equation to get the matching y-values.
Interpreting the solutions
Geometrically these solutions are the intersection points of a line with a curve, so there can be two, one, or no real solution pairs. Each x-value pairs with exactly one y-value from the linear equation — never mix them up. The number of solutions matches the discriminant of the quadratic you obtain: two distinct roots give two crossing points, a repeated root means the line is a tangent to the curve, and a negative discriminant means the line and curve never meet. Always substitute using the linear equation to recover y, since it gives a single value cleanly, whereas the curve could give two.
Key formula
From the linear equation write y = (a subject), substitute into the non-linear equation to obtain Ax² + Bx + C = 0, solve, then find y for each x. Keep x and y paired.
Worked example
Solve y = x + 1 and y = x² − 1 simultaneously.
Substitute: x² − 1 = x + 1 ⇒ x² − x − 2 = 0 ⇒ (x − 2)(x + 1) = 0 ⇒ x = 2 or x = −1.
Then y = x + 1: when x = 2, y = 3; when x = −1, y = 0. Solutions: (2, 3) and (−1, 0).
Remember
- Substitute the linear equation into the non-linear one, not the reverse.
- Each x-value has its own y-value — pair them correctly.
- A repeated root means the line is a tangent to the curve (one solution).