A line and a nonlinear equation

A simultaneous solution must satisfy both equations. Use the linear equation to express one variable in terms of the other; substitution turns the nonlinear equation into one variable. Each accepted root gives a paired value for the other variable.

Example

For \(y=x+1\) and \(x^2+y^2=13\), substitute to obtain \(x^2+(x+1)^2=13\), so \(2x^2+2x-12=0\). Thus \((x+3)(x-2)=0\).

If \(x=2\), then \(y=3\). If \(x=-3\), then \(y=-2\). The two solutions are \((2,3)\) and \((-3,-2)\); both give \(x^2+y^2=13\).

Do not mix an x-value from one root with the y-value from the other. A context can reject a negative length, but an unrestricted coordinate need not be positive.