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