#### Answer

Number of goats = 14
Number of chickens = 9

#### Work Step by Step

Let the number of goats be x and the number of chickens be y.
Since we know the total number of animals is 23, we can say:
$x+y=23$
Since each goat has 4 legs and each chicken has 2 legs, we can also get
$4x+2y=74$
To get a common term in both equations, we can multiply both sides of the first equation by 2 and distribute:
$2(x+y)=2(23)$
$2x+2y=46$
Now that both equations have a common term (2y), we can find the difference and then solve for x:
$(4x+2y)-(2x+2y)=(74)-(46)$
$4x+2y-2x-2y=28$
$2x=28$
$x=14$
We can plug this value back into the first equation to find y:
$(14)+y=23$
$y=9$
Therefore:
Number of goats = 14
Number of chickens = 9