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

To determine the condition for a system of linear equations to have a unique solution, we consider the following pair of linear equations:

\(px + qy = r\)

\(p_1x + q_1y = r_1\)

The system of equations will have a unique solution if the determinant of the coefficients matrix is non-zero. For a system of two equations, this condition can be written as:

\(\text{Det} = \begin{vmatrix} p & q \\ p_1 & q_1 \end{vmatrix} \neq 0\)

The determinant is calculated as:

\(pq_1 - qp_1\)

Therefore, for the system of equations to have a unique solution, the following condition must be satisfied:

\(pq_1 \neq qp_1\)

Let's analyze the given options:

1. \(pq \neq p_1q_1\): This condition is incorrect for ensuring a unique solution as it involves incorrect terms from the determinant matrix.
2. \(pp_1 \neq qq_1\): This is again an incorrect form, looking at the absolute magnitudes, unrelated to the determinant of coefficients.
3. \(pq_1 \neq qp_1\): This option represents precisely the condition derived from the determinant being non-zero. Thus, this is the correct condition for a unique solution.
4. \(pqr \neq p_1q_1r_1\): This condition involves the constants \(r\) and \(r_1\), which are not relevant for determining the determinant condition for a unique solution.

Hence, the correct answer is:

\(pq_1 \neq qp_1\)