Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Let \(a, b, c \in \mathbb{R}\). Consider the system of linear equations<br/>\(ax + 2y = b\)<br/>\(3x + 2y = c\)<br/>Which of the following statement(s) is (are) correct?</p>
<p>(a) If \(a = -3\), then the system has infinitely many solutions</p>
Step-by-Step Solution
Key Concept: Analyze the coefficient determinant and consistency conditions for systems with parameters.
<p><strong>Solution:</strong> If $a = -3$, the system becomes:<br/>$-3x + 2y = b$<br/>$3x + 2y = c$<br/>Adding these equations: $4y = b + c$. For infinitely many solutions, we need $b = -c$. The coefficient matrix has determinant $a \cdot 2 - 2 \cdot 3 = 2a - 6 = 2(-3) - 6 = -12 ≠ 0$, so there is a unique solution unless the system is inconsistent.</p>
Correct Answer: A