Solve the following simultaneous equations: 27/(x - 2) + 31/(y + 3) = 85 ; 31/(x - 2) + 27/(y + 3) = 89

Answer: \((x, y) = \left(\frac{5}{2}, -2\right)\)

Put \(\frac{1}{x-2} = m\) and \(\frac{1}{y+3} = n\): \(27m + 31n = 85\) ... (I), \(31m + 27n = 89\) ... (II).

Adding (I) and (II):\[58m + 58n = 174\]\[m + n = 3 \quad \text{(III)}\]

Subtracting (I) from (II):\[4m - 4n = 4\]\[m - n = 1 \quad \text{(IV)}\]

Adding (III) and (IV): \(2m = 4\), \(m = 2\); so \(n = 1\).

\(\frac{1}{x-2} = 2\) gives \(x - 2 = \frac{1}{2}\), so \(x = \frac{5}{2}\).
\(\frac{1}{y+3} = 1\) gives \(y + 3 = 1\), so \(y = -2\).