Answer:
There are 12 horses and 18 geese.
Step-by-step explanation:
We are given that in a field full of horses and geese, a farmer notes that there are 30 heads and 84 feet.
We can write a system of equations using the given information.
Let the amount of horses there are be represented by h and geese by g.
Assuming each horse and geese has only one head, we can write that:
[tex]\displaystyle h + g= 30[/tex]
And assuming that each horse has four feet and each geese has two feet, we can write that:
[tex]4h + 2g = 84[/tex]
This yields a system of equations:
[tex]\displaystyle \left\{\begin{array}{l} h + g = 30 \\ 4h + 2g = 84 \end{array}[/tex]
We can solve it using substitution. From the first equation, isolate either variable:
[tex]g = 30 - h[/tex]
From the second, we can first divide by two:
[tex]2h + g = 42[/tex]
And substitute:
[tex]2h + (30 - h) = 42[/tex]
Combine like terms:
[tex]h + 30 = 42[/tex]
Subtract:
[tex]h = 12[/tex]
Therefore, there are 12 horses.
And since the total number of animals is 30, there must be 30 - 12 or 18 geese.
In conclusion, there are 12 horses and 18 geese.
