Let's solve a generic 2x2 system of linear equations. 

To solve for x, let's eliminate the y's by multiplying the
top equation by d and the bottom equation by b 

When you add the two equations together, and solve for x,
you get 

Now, let's solve for y by eliminating the x's. Multiply
the top equation by c and the bottom equation by a. 

When you add the two equations together, and solve for y,
you get 

Now, consider the following definitions 
The determinant D formed by taking the coefficients 

The determinant D_{x} formed by taking the coefficient
matrix and replacing the x's by the constants on the right hand side. 

The determinant D_{y} formed by taking the coefficient
matrix and replacing the y's by the constants on the right hand side. 

Have you seen those determinants anywhere before? If not,
then you've not been reading the lecture notes. 
