Math Problem Solver

The system of equations has \(2\) solutions.

\[x = -2, y = 4\]\[x = 3, y = 9\]

Solve \(y = x + 6\) for \(x\). \[x = y - 6\]Substitute \(y - 6\) for \(x\) in \(y = x^{2}\) and simplify. $$\begin{aligned}y &= x^{2} \\ y &= \left(y - 6\right)^{2} \\ y^{2} - 13 y &= -36 \\y^{2} - 13 y + 36 &= 0 \\ \left(y - 9\right) \left(y - 4\right) &= 0 \\ y = 4&, y = 9\end{aligned}$$Substitute \(4\) into \(y = x + 6\) to solve for \(x\). \[\begin{aligned}4 &= x + 6\\- x &= 2\\x &= -2\end{aligned}\]This yields the following solution. $$\begin{aligned}x = -2,\,y = 4\end{aligned}$$Substitute \(9\) into \(y = x + 6\) to solve for \(x\). \[\begin{aligned}9 &= x + 6\\- x &= -3\\x &= 3\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 3,\,y = 9\end{aligned}$$