2(a, 3, c) + 3(c, 7, c) = (5, b +c, 15)
(2a, 6, 2c) + (3c, 21, 3c) = (5, b +c, 15)
(2a+3c, 27, 5c) = (5, b +c, 15)

Equating the individual elements of the two column vectors
2a + 3c = 5 _______ (1)
b + c = 27 _______ (2)
5c = 15 _____ (3)

From (3), c = 3.
Putting c = 3 in (2), b = 24
Putting c = 3 in (1), a = -2