Simultaneous equations (linear and quadratic)

What simultaneous equations are

• Simultaneous equations are two equations with two unknowns (usually $x$ and $y$) that must both be true at once.
• The solution is the pair (or pairs) $(x, y)$ that satisfies both equations.
• Graphically, each equation is a curve or line, and the solutions are the points where they meet.
• At Level 2 you handle two linear equations, and — the main new skill — one linear and one quadratic.

Two linear equations

• Eliminate one unknown by adding or subtracting the equations, or substitute one into the other.
• The solution is the single point where the two lines cross.

One linear and one quadratic

• Rearrange the linear equation to make one variable the subject (this is the easy one to move).
• Substitute it into the quadratic to get a single quadratic equation in one unknown.
• Solve that quadratic (factorise or use the formula).
• Find the matching value of the other variable for each solution, by substituting back into the linear equation.

Interpreting the solutions

• A line and a parabola can meet at two points, one point (the line is a tangent), or not at all.
• The number of meeting points equals the number of real solutions of the quadratic — so the discriminant tells you which case you are in.
• Each solution pair is a point of intersection $(x, y)$.