Solve the following simultaneous equations: 2/x - 3/y = 15 ; 8/x + 5/y = 77

Answer: \((x, y) = \left(\frac{1}{9}, 1\right)\)

Put \(\frac{1}{x} = m\) and \(\frac{1}{y} = n\).

The equations become \(2m - 3n = 15\) ... (III) and \(8m + 5n = 77\) ... (IV).

Multiply (III) by 5: \(10m - 15n = 75\)
Multiply (IV) by 3: \(24m + 15n = 231\)

Add the equations to eliminate \(n\):\[34m = 306\]\[m = 9\]

Substitute \(m = 9\) in (III): \(-3n = 15 - 18 = -3\), so \(n = 1\).

Check in (IV): \(8(9) + 5(1) = 77\), which is true.

\(m = \frac{1}{x} = 9\), so \(x = \frac{1}{9}\); \(n = \frac{1}{y} = 1\), so \(y = 1\).