A system of equations does not have a unique solution when the equations are not linearly independent. The equations are linearly independent if none of the left-hand side of the equations can be represented as a linear combination of the others.
If we add equations (1) and (3)
-1x + 7y + 2z = 6 ...(1)
3x - 6y - 2z = -25 ...(3)
we get
2x +1y + 0z = -19 ...(2A)
Examine the left-hand side of equation (2A) carefully.
What value of k in equation (2) would make its left-hand side identical to a linear combination of (1) and (3), namely equation (2A)?
For what value of k does the system below not have a unique solution?
-1x + 7y + 2z = 6
2x + 1y + kz = -11
3x - 6y - 2z = -25
A.5
B.0
C.-2
D.-8
1 answer