Afarmer notes that in a field of goats and geese there are 41 heads and 114 feet how many of each animal are there

There are 25 geese and 16 goats

Step-by-step explanation:

Let O represent the # of goats and E the # of geese.  Remembering that goats have four feet and geese only two, and that each animal has only one head, we can write the following equations to determine O and E:

Number of heads:  41; this is the total of the # of goat heads and the # of goose heads:  O + E = 41

(4 feet per goat)O + (2 feet per goose)E = 114 (There are a total of 114 feet)

Then we solve the following system of linear equations:

  O   +   E   =    41

 4O   +   2E  = 114

Multiplying the first equation by -2 results in -2O  - 2E  = -82.

Combining this result with the second equation, we obtain the system

 4O  +  2E  =  114

 -2O  - 2E  = -82

  2O =   32

Then O = # of goats = 32/2, or 16, and so the equation O   +   E   =    41 becomes

         E  = # of geese =  41 - O (remembering that we found O to be 16), or

         E  =  41 - 16  = 25

There are 25 geese and 16 goats.