# How do you solve the following system:  5x+y=-7 , 12x-9y=-3 ?

Feb 7, 2016

$x = - \frac{22}{19}$, $y = - \frac{23}{19}$.

#### Explanation:

I would recommend the method of elimination.

We have our 2 equations:

$5 x + y = - 7$
$12 x - 9 y = - 3$

Take the first equation and multiply it through by $9$ to obtain, this will allow us to get the same number of $y$s on both equations so we can add them and eliminate as follows

$45 x + 9 y = - 63$

We can now add this to the second equation and we get:

$\left(12 x - 9 y\right) + \left(45 x + 9 y\right) = \left(- 3\right) + \left(- 63\right)$

Now, by gathering the like terms we see that $y$ cancels to $0$.

$57 x = - 66 \to x = - \frac{66}{57} = - \frac{22}{19}$

Now that we have a value for $x$ put this value into back into either of the first or second equation and solve for $y$. Here we will use the first equation and get:

$5 \left(- \frac{22}{19}\right) + y = - 7$
$\to y = 5 \left(\frac{22}{19}\right) - 7 = \frac{110}{19} - \frac{133}{19} = - \frac{23}{19}$

And so we see that:

$x = - \frac{22}{19}$, $y = - \frac{23}{19}$.

As we chose the first equation to put our value of $x$ into it is good practice to check these to make sure that the second equation is satisfied as well.

$12 \left(- \frac{22}{19}\right) - 9 \left(- \frac{23}{19}\right) = - \frac{264}{19} + \frac{207}{19} = - \frac{57}{19} = - 3$

So the second equation is also satisfied.