Vector help, please!

Question

I feel like the answer is right in front of me. I just keep arriving at the same solution which is apparently the wrong one. Could someone guide me to an easier way of approaching this problem that maybe I'm not seeing? Intermediate steps are always helpful!

 

Three vectors A, B, and C are related to one another such that 3A − 5B = 30C.
A fourth vector D exists such that 15C + D = A.
Determine an expression for D in terms of the vectors A and B.

Answer 1

3A - 5B = 30C

15C + D = A  ==>  D = A - 15C  

 

Divide the 1st equation by 2 to get  15C = (3/2)A - (5/2)B  

and substitute this value for 15C in the 2nd equation  

 

D = A - [(3/2)A - (5/2)B]  

D =  -(1/2)A + (5/2)B  

 

As a check, note that since 15C + D = A it must be true that that 30C + 2D = 2A, and  

 

30C + 2D = (3A - 5B) + 2[-(1/2)A + (5/2)B] = 3A - 5B - A + 5B = 2A