2.(a, 3, c) + 3(c, 7, c) = (5, b+c, 15)
Treat the above as a vector equation.
2.(a, 3, c) + 3(c, 7, c) = (5, b+c, 15) multiply in the two factors
(2a, 6, 2c) + (3c, 21, 3c) = (5, b+c, 15) add the 1st two vectors
(2a + 3c, 6 + 21, 2c + 3c) = (5, b + c, 15)
(2a + 3c, 27, 5c) = (5, b + c, 15) now, match corresponding elements
2a + 3c = 5 -------- (1)
27 = b + c ---------- (2)
5c = 15 ------------- (3)
From (3), c = 5. Substitute for c = 5 in (1) and (2), giving
2a + 15 = 5 -------- (4)
27 = b + 5 ----------- (5)
From (5), b = 22. From (4), a = -5.
Answer: a = -5, b = 22, c = 5