# How do I use Cramer's rule to solve the system of equations 3x+4y=19 and 2x-y=9?

Mar 1, 2015

$x = 5$
$y = 1$.

Remembering that the derminant of a matrix $2 \times 2$,

$a$ $b$
$c$ $d$

is: $\Delta = a d - b c$

First of all let's write the matrix of the coefficients:

$3$ $4$
$2$ $- 1$

and let's find the determinant: $\Delta = 3 \cdot \left(- 1\right) - 2 \cdot 4 = - 3 - 8 = - 11$, that is not zero, so the system is possible.

Now let's find ${\Delta}_{x}$, that is the determinant of the previous matrix, in which we have to substitue the column $x$ (the first) with the coloumn of the known terms:

$19$ $4$
$9$ $- 1$

so: ${\Delta}_{x} = 19 \cdot \left(- 1\right) - 9 \cdot 4 = - 19 - 36 = - 55$.

And now the same thing with the second column:

$3$ $19$
$2$ $9$

so: ${\Delta}_{y} = 3 \cdot 9 - 2 \cdot 19 = 27 - 38 = - 11$.

And now the solution is:

$x = {\Delta}_{x} / \Delta = \frac{- 55}{-} 11 = 5$

$y = {\Delta}_{y} / \Delta = \frac{- 11}{-} 11 = 1$.