Answer
l=$\frac{c}{a}$; 0x + y(d-$\frac{bc}{a}$)=(g-$\frac{cf}{a}$); y=$\frac{ag-cf}{ad-bc}$
Work Step by Step
Start by creating a matrix from the given system of equations. Our goal is to create a triangular system of equations so we can solve for y.
{a b | f
c d | g}
By multiplying the first row vector by $\frac{c}{a}$, and using the formula $R_{2}$-$R_{1}$=$R_{2'}$
{c d | g} - {c $\frac{bc}{a}$ | $\frac{cf}{a}$} = {0 (d-$\frac{bc}{a}$) | (g-$\frac{cf}{a}$)}
Our new augmented matrix will be:
{a b | f
0 (d-$\frac{bc}{a}$) | (g-$\frac{cf}{a}$)}
Now that we have a triangular matrix, we can convert it back into a system of equations. Our new system of equations will be:
ax+by=f
y(d-$\frac{bc}{a}$)=(g-$\frac{cf}{a}$)
We can now solve for y by dividing both sides of the second equation by (d-$\frac{bc}{a}$):
y=(g-$\frac{cf}{a}$)$\div($d-$\frac{bc}{a}$)
y=($\frac{ag-cf}{a}$)$\div$($\frac{ad-bc}{a}$)
y=($\frac{ag-cf}{a}$)$\times$($\frac{a}{ad-bc}$)
y=$\frac{ag-cf}{ad-bc}$
