Mathematics

Problem

The following system of equations does not have a unique solution.

\begin{cases} 6x + ky + 2z = a \\ 6x - y - z = 7 \\ 2x - 3y + z = 1 \end{cases}

(a) Find the value of

k

. b. Each equation represents a plane. (b) Find (i) the value of

a

for which the three planes intersect in a line (ii) the equation of the line. (c) If the value of

a

is such that the three planes do not intersect in a line, describe their geometric configuration, justifying your answer.

Intuitive explanation

The key insight is that when a system lacks a unique solution, the equations are dependent — one row is a linear combination of the others, or they're inconsistent. Start by using the second and third equations (which don't involve the unknowns $k$ and $a$ ) to find a relationship between $x$ , $y$ , and $z$ . Then, the first equation must be compatible with this constraint: either it's a multiple of a combination of the other two (infinitely many solutions) or it contradicts them (no solution). Your job is to find which values of $k$ and $a$ make the system dependent rather than uniquely solvable.

Intuitive explanation

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