3x+2y=2xy,6x+2y=3xy

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on Mar 16, 2025 at 8:41 pm



You asked:

Solve the system of equations:3x+2y=2xy6x+2y=3xy\begin{aligned}3 x + 2 y &= 2 x y\\6 x + 2 y &= 3 x y\end{aligned}

MathBot Answer:

The system of equations has 22 solutions.

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


Solve 3x+2y=2xy3 x + 2 y = 2 x y for xx. x=2y2y3x = \frac{2 y}{2 y - 3}Substitute 2y2y3\frac{2 y}{2 y - 3} for xx in 6x+2y=3xy6 x + 2 y = 3 x y and simplify. 6x+2y=3xy6(2y2y3)+2y=3(2y2y3)y6y22y3=2y(2y+3)2y3\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 00 into 3x+2y=2xy3 x + 2 y = 2 x y to solve for xx. 3x=0x=0\begin{aligned}3 x &= 0\\x &= 0\end{aligned}This yields the following solution. x=0,y=0\begin{aligned}x = 0,\,y = 0\end{aligned}Substitute 33 into 3x+2y=2xy3 x + 2 y = 2 x y to solve for xx. 3x+6=6x3x=6x=2\begin{aligned}3 x + 6 &= 6 x\\- 3 x &= -6\\x &= 2\end{aligned}This yields the following solution. x=2,y=3\begin{aligned}x = 2,\,y = 3\end{aligned}

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