The number of dogs and chickens add up to 14. The number of legs between them is 36. How many dogs and how many chickens are on the farm if there are at least twice as many chickens as dogs?
To solve this problem, we can use a system of equations. Let's assume the number of dogs is x, and the number of chickens is y.
From the information given, we know that:
1. The number of dogs and chickens add up to 14: x + y = 14.
2. The number of legs between them is 36: 4x + 2y = 36 (since each dog has 4 legs and each chicken has 2 legs).
Now, we need to incorporate the fact that there are at least twice as many chickens as dogs. Mathematically, this can be expressed as y ≥ 2x.
To solve this system of equations, we can use the substitution method.
Step 1: Solve the first equation for x in terms of y.
x = 14 - y
Step 2: Substitute this value of x into the second equation.
4(14 - y) + 2y = 36
Step 3: Simplify and solve for y.
56 - 4y + 2y = 36
-2y = 36 - 56
-2y = -20
y = -20 / -2
y = 10
Step 4: Substitute the value of y back into the first equation to find x.
x + 10 = 14
x = 14 - 10
x = 4
Therefore, there are 4 dogs and 10 chickens on the farm.