Math Problem Solver

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

\[x = 6, y = 8\]\[x = 8, y = 4\]

Solve \(y = 20 - 2 x\) for \(x\). \[x = 10 - \frac{y}{2}\]Substitute \(10 - \frac{y}{2}\) for \(x\) in \(y = x^{2} - 16 x + 68\) and simplify. $$\begin{aligned}y &amp= x^{2} - 16 x + 68 \\ y &= \left(10 - \frac{y}{2}\right)^{2} - 16 \left(10 - \frac{y}{2}\right) + 68 \\ y^{2} - 12 y &= -32 \\y^{2} - 12 y + 32 &= 0 \\ \left(y - 8\right) \left(y - 4\right) &= 0 \\ y = 4&, y = 8\end{aligned}$$Substitute \(4\) into \(y = 20 - 2 x\) to solve for \(x\). \[\begin{aligned}4 &= 20 - 2 x\\2 x &= 16\\x &= 8\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 8,\,y = 4\end{aligned}$$Substitute \(8\) into \(y = 20 - 2 x\) to solve for \(x\). \[\begin{aligned}8 &= 20 - 2 x\\2 x &= 12\\x &= 6\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 6,\,y = 8\end{aligned}$$