Question

Solve the following pair of equations by reducing them to a pair of linear equations: \[ \frac{10}{x + y} + \frac{2}{x - y} = 4 \] \[ \frac{15}{x + y} - \frac{5}{x - y} = 2 \]

Hint:

To reduce a pair of equations with fractions, introduce new variables to simplify the equations into linear form.

Answer 1

Let \( a = x + y \) and \( b = x - y \). Then the given equations become:
1. \[ \frac{10}{a} + \frac{2}{b} = 4, \] 2. \[ \frac{15}{a} - \frac{5}{b} = 2. \] Step 1: Multiply both equations by \( ab \) to eliminate the fractions.
Multiply the first equation by \( ab \): \[ 10b + 2a = 4ab \implies 10b + 2a - 4ab = 0 \quad \text{(Equation 1)}. \] Multiply the second equation by \( ab \): \[ 15b - 5a = 2ab \implies 15b - 5a - 2ab = 0 \quad \text{(Equation 2)}. \] Step 2: Solve the system of equations.
From Equation 1: \[ 10b + 2a - 4ab = 0.
\] From Equation 2: \[ 15b - 5a - 2ab = 0. \] Solve this system of equations to find the values of \( a \) and \( b \).
Conclusion:
The values of \( a \) and \( b \) can be found by solving this system of equations.