The solution of $2/x+3/y=13$ and $5/x-4/y=-2$, where $x$ and $y$ are non-zero, is:

Put $u=1/x$ and $v=1/y$, so $2u+3v=13$ and $5u-4v=-2$. Multiplying by 4 and 3 gives $8u+12v=52$ and $15u-12v=-6$; adding, $23u=46$ so $u=2$ and then $v=3$. Hence $x=1/u=1/2$ and $y=1/v=1/3$. Check: 4 + 9 = 13 and 10 − 12 = −2.