Discussion
For the following set of consistent equations what is the value of constant K ?
x - y + z = 1
2x + 2y - z = 2
3x - y + 2z = 3
4x + y + z = K
Answer: D
The constant is K = 4.
Solve the first three equations, then require the fourth to agree. The coefficient determinant of the first three is
\[ \begin{vmatrix} 1 & -1 & 1 \\ 2 & 2 & -1 \\ 3 & -1 & 2 \end{vmatrix} = 3 + 7 - 8 = 2 \]
which is not zero, so they have exactly one solution. By inspection \(x = 1, y = 0, z = 0\) satisfies all three (1 = 1, 2 = 2, 3 = 3). For the system to be consistent the fourth equation must hold at the same point:
\[ K = 4(1) + 0 + 0 = 4 \]
Any other value, such as 6, 8 or 12, would make the fourth equation contradict the first three.
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