Solve the following linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent.

Question

2x - 3y + 4z = 0

x - y - 2z = 0

x- 3y + 14z = 0

Answer 1

The determinant is 0, so we have dependent solutions. Since the right hand side is 0 in all cases, the solutions will be consistent. (If not 0, we would have to verify that the right hand side values yield consistent solutions.)

If we wish to derive the solution, take the first 2 rows and calculate the cross product

i j k

2 -3 4

1 -3 -2

= 10i + 8j + k

The solution is t(10, 8, 1) If you wish, you can verify that this is the solution by plugging in to the three equations with t = 1 and seeing that we get 0 for each right hand side.

Answer 2

x=y=z = 0

(x,y,z) = (0,0,0)

independent, consistent