Question

If \( \frac{5}{x+1} + \frac{1}{y-3} = 2 \) and \( \frac{6}{x+1} - \frac{3}{y-3} = 1 \), then \( x = \dots \):

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

Substitute new variables to simplify systems involving fractions. It can make the algebra easier to handle.

Answer 1

The Correct Option is B

Step 1: Let \( p = x + 1 \) and \( q = y - 3 \).
This substitution simplifies the system of equations: \[ \frac{5}{p} + \frac{1}{q} = 2 \quad \text{(1)} \] \[ \frac{6}{p} - \frac{3}{q} = 1 \quad \text{(2)} \] Step 2: Multiply equation (1) by \( p \) and equation (2) by \( q \).
Multiplying equation (1) by \( p \) and equation (2) by \( q \), we obtain: \[ 5q + p = 2pq \quad \text{(3)} \] \[ 6q - 3p = pq \quad \text{(4)} \] Step 3: Solve the system.
Now, solve equations (3) and (4) to find the values of \( p \) and \( q \). After solving, we find \( x = 2 \).