Simultaneous linear and quadratic equations

For two linear equations, elimination can remove one unknown by adding suitable multiples. For a linear equation and a quadratic relation, solve the linear equation for one variable and substitute it into the quadratic. The resulting quadratic may give zero, one or two real intersections.

Each root belongs to its own ordered pair. Substitute every candidate into the linear equation to find its matching second coordinate, then check both original equations. Do not mix the x-value of one solution with the y-value of another. A graphical sketch can explain the number of intersections, while algebra supplies exact coordinates.

Worked example

Solve y = x + 1 and y = x² − 1.

1. Substitute: x + 1 = x² − 1, so x² − x − 2 = 0.
2. Factor (x − 2)(x + 1) = 0, giving x = 2 or x = −1.
3. Use y = x + 1 for each value separately.

Answer: (x,y) = (2,3) or (−1,0)

Revise first: Solving quadratic equations.