How does determinant method of solving equations work? Say you have two equations

$ax+by+c=0$

$dx+ey+f=0$

by determinant method

$\frac{x}{bf-ce}=\frac{-y}{af-cd}=\frac{1}{ae-db}$

Clearly, the solution is valid

How exactly was this derived?
Anvit, using my set up of equations, here is a way: 
Multiply $ax+by=e$ by $d$ to get $adx+bdy=ed$ and multiply $cx+dy=f$ by $b$ to get $bcx+bdy=bf$. Now eliminating $bdy$ from both equations through subtraction yields $(ad-bc)x=ed-bf$ from which $x$ follows by division $ad-bc$ which is called determinant. For $y$ you can do the same thing. Can you give this a try?

EDIT: Considering $ad-bc$ and $ed-bf$ as determinants of two matrices, it ultimately gives Cramer's Rule
it is $$x=\frac{ce-bf}{bd-ae},y=\frac{cd-af}{ae-bd}$$
and $$bd-ae$$ is the determinant, see also the Cramers rule