Math Problem Solver

The system of equations has $2$ solutions.

$$x = 0, y = 0$$$$x = 2, y = 3$$

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