Question

Farmer John has cows and chickens on his farm. His farm animals have 120 legs and 48 heads total. How many cows and how many chickens are on the farm?

a. Explain what 120 represents, and how it relates to the cows and chickens.

b. Explain what 48 represents, and how it relates to the cows and chickens.

c. Setup a system, of two equations, to help you solve this riddle.

d. Solve this system using either Elimination or Substitution, show your work, and state your answer as a complete sentence.​

Answer 1

Using a system of equations representing the number of legs and heads, we determine that Farmer John has 12 cows and 36 chickens on his farm based on the provided totals of 120 legs and 48 heads.

Step-by-step explanation:

Farmer John has cows and chickens on his farm. We're tasked to determine the number of each animal given the total number of legs and heads. The number 120 legs represents the total legs of cows and chickens combined, with cows having 4 legs each and chickens having 2. The number 48 heads simply represents the total number of animals since each animal has one head.

To set up a system of equations, we denote the number of cows as c and the number of chickens as h. Since each cow has 4 legs and each chicken has 2, and together they have 120 legs, our first equation is 4c + 2h = 120. Since there are 48 animals in total, our second equation is c + h = 48.

Solving the system using substitution, we first solve the second equation for c, getting c = 48 - h. Substituting this into the first equation gives us 4(48 - h) + 2h = 120. Simplifying, we get 192 - 4h + 2h = 120, which reduces to 2h = 72. Therefore, h = 36. Substituting h back into c = 48 - h gives us c = 48 - 36 = 12. Consequently, Farmer John has 12 cows and 36 chickens on his farm.

Answer 2

a. 120 represents the total number of cow legs and chicken legs

b. 48 represents the total number of cow heads and chicken heads

c. System:

Let x = number of cows

Let y = number of chickens

1. 2x + 4y = 120

2. x + y = 48

d. I am going to use substitution for this problem

Step 1: Solve for y in the equation x + y = 48

1. x + y = 48 → y = -x + 48

Step 2: Substitute -x + 48 for y in 2x + 4y = 120

2. 2x + 4(-x + 48) = 120

Step 3: Solve for x

3. 2x - 4x + 184 = 120 → -2x + 184 = 120 → -2x = -64 → x = 32

Step 4: Sustitute 32 for x in x + y = 48

4. (32) + y = 48

Step 5: Solve for y

5. y = 16

Number of chickens on the farm (y) = 16

Number of cows on the farm (x) = 32

Step-by-step explanation:

All of it is above :)