I would recommend the method of elimination.

We have our 2 equations:

#5x+y = -7#

#12x-9y = -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

#45x+9y = -63#

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

#(12x-9y)+(45x+9y) = (-3) + (-63) #

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

#57x = -66 -> x = -66/57=-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(-22/19)+y=-7 #

#-> y = 5(22/19)-7=110/19 - 133/19=-23/19#

And so we see that:

#x=-22/19#, #y=-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(-22/19) -9(-23/19) =-264/19+207/19=-57/19=-3#

So the second equation is also satisfied.