Let r be the number of red spheres, w be the number of whites, and b the number of blues. We have
r+b=6w and r+2w=3b ... What you need to do is to get rid of the b. Here's what I did. Rewrite the equations as
r - 6w + b = 0
r + 2w - 3b = 0
Multiply the top equation by 3 and add the two equations togeter.
3r - 18w + 3b = 0
r + 2x - 3b = 0
4 r - 16 w = 0
Can you take it from there?
This is just a matter of practice. You'll get it.