Solve using the elimination method.
2r - 5s = -14
5r + 2s = 23

Since you can't multiply either equation by a single number to make the r's or s's cancel out you need to multiply both equations. 

I'm going to make the r's cancel out by multiplying the top equation by 5 and the bottom by -2

 5(2r - 5s  = -14)
-2( 5r + 2s = 23)

Once I multiply I have

  10r -25s = - 70
-10r - 4s = -46

Now add straight down, and you are left with:

-29s = -116

To get s by itself now you need to divide both sides by -29

-29s = -116
-29         -29

And your result is s = 4

To find the value of r, substitute s = 4 into one of the original equations and solve for r.  

2r - 5(4) = -14 

2r - 20 = -14
      + 20    + 20

2r = 6

so r = 3

So your solution is s=4 and r =3