Linearly Independent Sets; Bases

Question

Please tell me how to solve below question

Consider the vectors u→= [-3, 10, 7], v→= [10, -6, 9], and w→= [38/3, -24, k].

What value of k will make the set {→u,→v,→w} linearly dependent?

You may submit your answer as a fraction, if necessary.

Answer 1

u, v and w are linearly dependent if you can express one in terms of the others.

You can verify that neither vector is a multiple of the other so the way this will work is if w = au + bv.

so equating components you have

-3a + 10b =38/3

10a - 6b = -24

7a + 9b = k

Use the first two to solve for a and b and substitute into the third equation to get the value of k