Math Problem Solver

The system of equations has one solution.

\[x = 1, y = 1\]

Solve \(3 x y + x + y = 5\) for \(x\). \[x = \frac{5 - y}{3 y + 1}\]Substitute \(\frac{5 - y}{3 y + 1}\) for \(x\) in \(4 x y + 2 x + 2 y = 8\) and simplify. $$\begin{aligned}4 x y + 2 x + 2 y &amp= 8 \\ 4 \left(\frac{5 - y}{3 y + 1}\right) y + 2 \left(\frac{5 - y}{3 y + 1}\right) + 2 y &= 8 \\ \frac{2 \left(y^{2} + 10 y + 5\right)}{3 y + 1} &= 8 \end{aligned}$$Substitute \(1\) into \(3 x y + x + y = 5\) to solve for \(x\). \[\begin{aligned}4 x + 1 &= 5\\4 x &= 4\\x &= 1\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 1,\,y = 1\end{aligned}$$