Try a matrix division:
% 7x+14y-6z=a
% 12x+5y+9z=b
% -5x+7y+15z=c
X = [7, 14, -6;
12, 5, 9;
-5, 7, 15]
Y = [11; 22; 33] % Whatever it may be.
% Solve for the coefficients, using a least squares approach.
coefficients = X \ Y
% Double check to see if we get back a, b, and c
a2 = 7 * coefficients(1) + 14 * coefficients(2) - 6 * coefficients(3)
b2 = 12 * coefficients(1) + 5 * coefficients(2) + 9 * coefficients(3)
c2 = -5 * coefficients(1) + 7 * coefficients(2) + 15 * coefficients(3)
