Need help with solving a linear system given certain limits.
Ok its been a while since my college math class days, so I may have lost a step and I may not explain this well, but here is my problem. I am a programmer trying to write an algorithm that will help me balance the difficulty of a game I am working on. Without getting into unnecessary details I have narrowed it down to this.
55a + 25b + 20c + 20d = A
25a + 65b + 20c + 20d = B
25a + 25b + 50c + 20d = C
25a + 25b + 20c + 70d = D
a >= 0
b >= 0
c >= 0
d >= 0
A, B, C, & D are constants I need to be able to enter as parameters, then solve the equations so that I get the minimum a+b+c+d.
Now using a solver I found online, I was able to find that without the >= 0 restrictions I can find the answer by: (sorry i couldn't get latex to work right)
a = (323A - 75B - 80C - 48D) / 12690
b = (-25A + 87B - 20C - 12D) / 4230
c = (-100A - 75B + 343C - 48D) / 12690
d = (-20A - 15B - 16C + 75D) / 4230
but that allows for negative values which I cannot have.
How would I find a set of solutions which includes the >= restrictions?