4x +3y = -45.5
-5x +8y = -43
multiply the top equation by 5
20x +3y = -227.5
-5x +8y = -43
multiply the bottom equation by 4
20x +3y = -227.5
-20x +32y = -172
add the equations
0x +35y = -399.5
35y = -799/800
y = (-799/800)*(1/35)
35 reduces to 5*7, but 799 doesn't reduce to anything with a 5 or 7 in it, so the whole thing won't reduce
y = -799/26000
look at the original equations
4x +3y = -45.5
-5x +8y = -43
multiply the top equation by 8
32x +24y = -364
-5x +8y = -43
multiply the bottom equation by -3
32x +24y = -364
15x -24y = 172
add the equations together
47x +0y = -192
47x = -192
x = -192/47
47 is prime and 47 does not go into 192 evenly so it doesn't reduce.
Answer: x = -192/47, y = -799/26000