Math Problem Solver

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

\[x = -3, y = 12\]\[x = \frac{1}{2}, y = \frac{17}{2}\]

Solve \(y = 9 - x\) for \(x\). \[x = 9 - y\]Substitute \(9 - y\) for \(x\) in \(y = 2 x^{2} + 4 x + 6\) and simplify. $$\begin{aligned}y &amp= 2 x^{2} + 4 x + 6 \\ y &= 2 \left(9 - y\right)^{2} + 4 \left(9 - y\right) + 6 \\ 2 y^{2} - 41 y &= -204 \\2 y^{2} - 41 y + 204 &= 0 \\ \left(y - 12\right) \left(2 y - 17\right) &= 0 \\ y = \frac{17}{2}&, y = 12\end{aligned}$$Substitute \(\frac{17}{2}\) into \(y = 9 - x\) to solve for \(x\). \[\begin{aligned}\frac{17}{2} &= 9 - x\\x &= \frac{1}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{1}{2},\,y = \frac{17}{2}\end{aligned}$$Substitute \(12\) into \(y = 9 - x\) to solve for \(x\). \[\begin{aligned}12 &= 9 - x\\x &= -3\end{aligned}\]This yields the following solution. $$\begin{aligned}x = -3,\,y = 12\end{aligned}$$