Is the vector x unique in the context of linear algebra?

[fk378]

Homework Statement

Determine if the vector x is unique.

x = (-15, -3, 0) + x_3 (10, 0, 1)

Note: The scalars should be vertically placed instead of horizontal.

The Attempt at a Solution

Seeing that there is a free variable, I said that x is not unique, but my teacher marked it wrong. Why is it unique, and in what case would it not be unique?

 

[Mathdope]

So, x_3 is the 3rd component of x correct? If that's the case (I don't see what else it could be) then I agree that it doesn't seem to be unique as we have

(1) x_1 = -15 + 10t
(2) x_2 = -3
(3) x_3 = t

Now t can be chosen to be any real number. Are you sure that you copied them correctly?