Question

Solve the following pair of equations: \[ \frac{3}{x} + \frac{2}{y} = 11, \quad \frac{4}{x} - \frac{5}{y} = 7. \]

Hint:

To solve systems of equations with fractions, substitute variables to simplify the expressions into linear equations.

Answer 1

We are given the pair of equations: 1. \( \frac{3}{x} + \frac{2}{y} = 11 \) 2. \( \frac{4}{x} - \frac{5}{y} = 7 \) Let us assume: \[ u = \frac{1}{x} \quad \text{and} \quad v = \frac{1}{y}. \] Thus, the equations become: 1. \( 3u + 2v = 11 \) 2. \( 4u - 5v = 7 \) Step 1: Solve the system of equations. Multiply the first equation by 5 and the second equation by 2 to eliminate \( v \): 1. \( 15u + 10v = 55 \) 2. \( 8u - 10v = 14 \) Add the two equations: \[ 15u + 10v + 8u - 10v = 55 + 14 \quad \implies \quad 23u = 69 \quad \implies \quad u = 3. \] Step 2: Substitute \( u = 3 \) into one of the original equations. Substitute \( u = 3 \) into \( 3u + 2v = 11 \): \[ 3(3) + 2v = 11 \quad \implies \quad 9 + 2v = 11 \quad \implies \quad 2v = 2 \quad \implies \quad v = 1. \] Step 3: Find \( x \) and \( y \). Since \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \), we have: \[ x = \frac{1}{u} = \frac{1}{3} \quad \text{and} \quad y = \frac{1}{v} = 1. \]
Conclusion:
The solution to the system of equations is \( x = \frac{1}{3} \) and \( y = 1 \).