Hi
t+u+v = 6a + 4b
4t+kv = (8+k)a + (12+2k)b
are parallel if
t+u+v and 4t+kv being parallel, there exists K such that t+u+v = K(4t+kv)
This means 6a + 4b = K [(8+k)a + (12+2k)b]
[6 - K(8+k)]a = [- 4 + K(12+2k)]b
a and b being independent (or supposed so) this leads to
6 - K(8+k) = 0 and - 4 + K(12+2k) = 0
K = 6/(8+k) and K = 4/(12+2k)
therefore 6/(8+k) = 4/(12+2k)