Question

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

Answer 1

The answer is:

`x=5`
`y=1`.

Remembering that the derminant of a matrix `2xx2`,

`a` `b`
`c` `d`

is: `Delta=ad-bc`

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

`3` `4`
`2` `-1`

and let's find the determinant: `Delta=3*(-1)-2*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*(-1)-9*4=-19-36=-55`.

And now the same thing with the second column:

`3` `19`
`2` `9`

so: `Delta_y=3*9-2*19=27-38=-11`.

And now the solution is:

`x=Delta_x/Delta=(-55)/-11=5`

`y=Delta_y/Delta=(-11)/-11=1`.