There will be no systematic way if you are not told which equations to use (this is true of real life problems but not often of homework problems), to get the desired overall equation.
For nearly every homework problem I've seen, it's easy enough figuring it out by inspection (of course, you want to be smart about this, rather than randomly trying combinations), that using some algorithm might actually be slower.
That said, what you want is simply the values of k_i such that
f(a,b,c,d,...,n) = \sum _i k_i f_i(a,b,c,d,...,n)
This simply results in a set of n simultaneous equations in the coefficients k_i, which may be solved by your chosen method (straight algebraic manipulation, using determinants, etc.)
Example (an easy one for simplicity, and I'm dispensing with details of the states, and such) :
We want CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O~, ~~\Delta H_{comb}
given
C + 2H_2 \rightarrow CH_4~,~~\Delta H_1
C + O_2 \rightarrow CO_2~,~~\Delta H_2
2H_2 + O_2 \rightarrow 2H_2O~,~~\Delta H_3
Assume that
\Delta H_{comb} = k_1 \Delta H_1 + k_2 \Delta H_2 + k_3 \Delta H_3
Then, comparing terms we have :
from~~CH_4~:~1 = -k_1~~~~~(1)
from~~H_2~:~0 = 2k_1 + 2k_3 ~~~~~(2)
from~~C~:~0 = k_1 + k_2 ~~~~~(3)
from~~O_2~:~2 = k_2 + k_3 ~~~~~(4)
from~~H_2O~:~-2 = -2k_3 ~~~~~(5)
from~~CO_2~:~-1 = -k_2 ~~~~~(6)
Notice how I use +ve signs for terms on the LHS and -ve signs for terms on the RHS.
Anyway, we have 6 equations with 3 unknowns, of which, we suspect the other 3 will be redundant (but we shall verify this).
(1) tells us that k_1 = -1. Substituting this in (3) gives k_2 = 1. And (5) gives k_3 = 1. Clearly, these values also satisfy (2) (4) and (6), so they are consistent.
Hence, \Delta H_{comb} = - \Delta H_1 + \Delta H_2 + \Delta H_3
Of course, this would have be easier done by inspection, but this shows that an approach exists that will work well for the more complex problems, perhaps.
