Solve 8/(x + y) + 6/(x - y) = 5 and 12/(x + y) + 4/(x - y) = 5.

Step 1. Change letters. With $u = \dfrac{1}{x + y}$ and $v = \dfrac{1}{x - y}$, the pair is $8u + 6v = 5$ and $12u + 4v = 5$.

Step 2. Match $u$. Multiply the first equation by $3$ and the second by $2$: $24u + 18v = 15$ and $24u + 8v = 10$.

Step 3. Subtract. $10v = 5$, so $v = \dfrac{1}{2}$. Then $8u + 3 = 5$ gives $u = \dfrac{1}{4}$.

Step 4. Back to $x$ and $y$. $x + y = 4$ and $x - y = 2$. Adding gives $2x = 6$, so $x = 3$ and $y = 1$.