Question

The system of linear equations \(px + qy = r\) and \(p_1x + q_1y = r_1\) has a unique solution, if :

• A. \(pq \neq p_1q_1\)
• B. \(pp_1 \neq qq_1\)
• C. \(pq_1 \neq qp_1\)
• D. \(pqr \neq p_1q_1r_1\)

Hint:

Think of the cross-product of the coefficients. If \(p q_1 - q p_1 = 0\), the lines are either parallel or coincident. If it's non-zero, they must intersect at exactly one point.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A system of two linear equations has a unique solution if the lines represent intersecting lines. Geometrically, this means their slopes are not equal.
Step 2: Key Formula or Approach:
For equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\), the condition for a unique solution is: \[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
Step 3: Detailed Explanation:
1. Here, \(a_1 = p, b_1 = q\) and \(a_2 = p_1, b_2 = q_1\).
2. The condition is:
\[ \frac{p}{p_1} \neq \frac{q}{q_1} \] 3. Cross-multiplying to remove the fractions:
\[ p q_1 \neq q p_1 \]
Step 4: Final Answer:
The condition for a unique solution is \(pq_1 \neq qp_1\).