Allied & ICAR Exam Bank • ALLIED_2628 NAT • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Solutions of linear equations

Question Statement:

Consider the system of equations: $x + y + z = 6$ $x + 2y + 3z = 10$ $x + 2y + k z = m$ If the system has infinitely many solutions, the value of $k$ must be:

Answer Key:

3

Step-by-Step Solution & Explanation

Row reducing $[A|b]$: $R_2 - R_1 \to [0, 1, 2 | 4]$ and $R_3 - R_2 \to [0, 0, k-3 | m-10]$. For infinitely many solutions, $k - 3 = 0 \implies k = 3$ (and $m = 10$).
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