GATE 2015 EC – Question 11
Consider a system of linear equations:
$x - 2y + 3z = -1$,
$x - 3y + 4z = 1$, and
$-2x + 4y - 6z = k$.
The value of $k$ for which the system has infinitely many solutions is ________.
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Show answer and explanation
Correct answer: 2
Explanation
The first two equations are independent and have a line of solutions. The third equation must be consistent with them, which happens when it is a multiple of the first. Since $-2(x - 2y + 3z) = -2x + 4y - 6z$ and the first right side is $-1$, we need $k = -2 \times (-1) = 2$.