Instead of trial and error, you could use algebra.
Horses and chickens, 48 heads, 182 legs
H + C = 48 (total animals is 48)
and
4H + 2C = 182 (equations from the definition of the problem)
C=48-H (basic algebra)
Therefore, plugging C back into the second equation gives:
4H + 96 -2H = 192
Subtract 96 from each side:
2H = 86
Divide both by 2:
H = 43
and so C has to equal 5.
5 Chickens, 43 Horses, and a simple application of algebra and the method of solving for 2 variables with 2 equations.
Q.E.D.