Matrices & Determinants
Infinite Solutions Condition
nta_pyq_2025_apr
Grade 12
Question:
If the system of linear equations $x + y + 2z = 6$, $2x + 3y + az = a + 1$, $-x - 3y + bz = 2b$ where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to:
Step-by-Step Solution
Key Concept: For infinitely many solutions, all four determinants $D, D_1, D_2, D_3$ must be zero. Set them equal to zero to obtain two equations in $a$ and $b$.
Setting $D = D_1 = D_2 = D_3 = 0$: from above relations, $a = -2, b = 10$. $7a + 3b = -14 + 30 = 16$.
Correct Answer: 16