Solving a system of equations with variables in denominator.

Question

Solve for $\{x,y,z\}$: \begin{cases}\dfrac1x+\dfrac2y-\dfrac1z=\dfrac43\\\\ \dfrac2x+\dfrac3y-\dfrac2z=\dfrac53\\\\ \dfrac3x+\dfrac4y-\dfrac6z=3 \end{cases}

My attempt: I have tried combining each equation like so:$$\frac {yz+2xz-xy}{xyz}=\frac43$$ for each equation, but I got nowhere.

Answer 1

Why don't you solve for $\frac1x$ and the like, and then just take the inverse?

Answer 2

Take $x_1=\frac1x, y_1=\frac1y, z_1=\frac1z$