On a farm, there are 18 heads and 134 legs. how many chickens and dogs are there? (50 pts.)

Chickens and dogs both have one head each. However, chickens have 2 legs each and dogs have 4 legs each. 

Knowing this, you can write two equations, one that adds up to the total number of heads and one that adds up to the total number of legs. Let's call the number of chickens "c" and the number of dogs "d." 

The equation for the number of heads is:
48 (total number of heads) = c + d (since each animal has one head each)

The equation for the number of legs is:
134 (total number of legs) = 2c (since there are 2 legs per chicken) + 4d (since there are 4 legs per dog)

Now you have two equations, 48 = c + d and 134 = 2c + 4d, with two variables you're trying to find. That means you can use a system of equations to solve for the value of c and d.

There are three ways of solving a system of equations: substitution, graphing, and elimination. I will be using substitution.
1) Solve one equation for the value of one variable. I will solve 48 = c + d for one variable because it's easier than solving the other equation for a variable, however it doesn't matter. I will also be solving it for c, but you can also solve it for d:


2) Plug the value of the variable you solved for into the other equation and solve for the value of the other variable. I will be plugging c = 48 - d into 134 = 2c + 4d and solving for d:


Since d, the number of dogs, equals 19, there are 19 dogs.

3) Finally, plug the value of the variable you just found (d=19) into one of the equations to find the value of the other variable. I'll be plugging d=19 into 48 = c + d because that will be easiest to solve for c, but you can plug it into the other equation:


Since c, the number of chickens, equals 29, there are 29 chickens.

There are 29 chickens and 19 dogs.

What’s the improper fraction?