How many integers satisfy the following equation
$$\frac{6}{x}+\frac{4}{y}=1$$
I tried combining and I get
$$6y+4x=xy$$
This gets confusing so I just tried trial and error and found three points $(-6,2),(2,-2),(3,-4),(-18,3)$ But this method seems a bit tedious.
By manipulating the equation, you get: \begin{equation} xy - 6y - 4x = 0 \end{equation} Then, by using Simon's Favorite Factoring Trick as @John Omielan mentioned in the comments, you can turn this equation into the form $(x-a)(y-b) = ab$, which is in our case: \begin{equation} (x-6)(y-4) = xy - 6y - 4x + 24 = 0 + 24 = 24 \end{equation} Then, you can list out the factors of $24$ in pairs. The table becomes: \begin{equation} \begin{array}{|c|c|} \hline \text{$x-6$} & \text{$y-4$} \\ \hline \pm24 & \pm1 \\ \hline \pm12 & \pm2 \\ \hline \pm8 & \pm3 \\ \hline \pm6 & \pm4 \\ \hline \pm4 & \pm6 \\ \hline \pm3 & \pm8 \\ \hline \pm2 & \pm12 \\ \hline \pm1 & \pm24 \\ \hline \end{array} \end{equation} Solving for $x,y$ in these factors would give you the list of solutions.
