Question

If \( p \) and \( q \) are numbers such that the pair of linear equations \( (p + 2)x + (q - 1)y = 10 \) and \( (q + 2)x + (p - 1)y = 10 \) have infinite solutions for \( x \) and \( y \), then \( p = q \).

• A. Always
• B. Sometimes
• C. Never
• D.
• E.

Hint:

For linear equations to have infinite solutions, the system must be consistent, and the coefficients of the variables must be proportional.

Answer 1

The Correct Option is A

For two linear equations to have infinite solutions, their coefficients must be proportional.
Step 1: Write down the two equations: \[ (p + 2)x + (q - 1)y = 10 \quad \text{(Equation 1)} \] \[ (q + 2)x + (p - 1)y = 10 \quad \text{(Equation 2)} \] Step 2: The condition for infinite solutions is that the ratios of the coefficients of \( x \), \( y \), and the constant term must be equal: \[ \frac{p + 2}{q + 2} = \frac{q - 1}{p - 1} = \frac{10}{10} = 1 \] Step 3: From \( \frac{p + 2}{q + 2} = 1 \), we get: \[ p + 2 = q + 2 \quad \Rightarrow \quad p = q \] Thus, for infinite solutions, it is always true that \( p = q \).