The legs add up, but the eyes do not. You are correct that there are 19 animals. But then you go on to finish wrong.
If there are m cows (moo) and c hens (cluck),
4m+2c = 52
2m+2c = 38
2m = 14
m = 7
7 cows and 12 chickens
7*4+12*2 = 28+24 = 52 legs
2*(7+12) = 2*19 = 38 eyes
Farmer Greg has some cows and some hens. Between them they have 52 legs and 38 eyes. How many hens, and how many cows does farmer Greg have?
Clue..
Cows have four legs
Hens have 2 legs
Eyes 2
What I got..
38÷2=19
8×4=32
10×2=20
20+32=52
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