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I've begun with the hint and found out that $(det\ A)*x=adj\ A*c$ and therefore what x is. My question would be how would I go about finding what $det\ A_i$ is? Should I go about it using the formal definition (summation of permutations)?
In case you'd want to know, this relation is known as Cramer's rule. Wikipedia and Cliffsnotes might provide some useful examples if you're interested.
You can use any method you like to come up with $det\ A_i$, for small matrices standard rules or cofactor expansion might work out fine, for larger matrices you can use row operations to get to a triangular matrix or at least a matrix with a bunch of zero entries as described here.
I've been trying to find a proper proof of Cramers's rule explicitly using the adjunct, but I can't seem to find it. The best proof I've actually found is outlined here, while the most promising sounding proof, Robinson, Stephen M. (1970). "A Short Proof of Cramer's Rule". Mathematics Magazine, sadly doesn't seem to be available online (or anywhere else really).