Answer :
### Step-by-Step Solution
Given the problem, we need to determine the number of sheep and chickens a farmer owns based on the total number of animals and legs.
#### Step 1: Set Up the System of Equations
Let's define our variables:
- [tex]\( x \)[/tex] = number of sheep
- [tex]\( y \)[/tex] = number of chickens
From the problem, we have the following two pieces of information:
1. The farmer has a total of 8 animals.
2. There is a total of 20 legs.
We can use this information to set up the following system of equations:
[tex]\[ \begin{cases} x + y = 8 \\ 4x + 2y = 20 \end{cases} \][/tex]
#### Step 2: Solve the System of Equations
Equation 1: [tex]\( x + y = 8 \)[/tex]
Equation 2: [tex]\( 4x + 2y = 20 \)[/tex]
Solve Equation 1 for [tex]\( y \)[/tex]:
[tex]\[ y = 8 - x \][/tex]
Substitute [tex]\( y = 8 - x \)[/tex] into Equation 2:
[tex]\[ 4x + 2(8 - x) = 20 \][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x + 16 - 2x = 20 \\ 2x + 16 = 20 \\ 2x = 4 \\ x = 2 \][/tex]
Now that we have [tex]\( x = 2 \)[/tex], substitute it back into the equation [tex]\( y = 8 - x \)[/tex]:
[tex]\[ y = 8 - 2 \\ y = 6 \][/tex]
Thus, the number of sheep is [tex]\( x = 2 \)[/tex] and the number of chickens is [tex]\( y = 6 \)[/tex].
#### Step 3: Verification
To ensure our solution is correct, we should verify it by plugging these values back into the original equations.
Verification with Equation 1:
[tex]\[ x + y = 2 + 6 = 8 \quad \text{(satisfied)} \][/tex]
Verification with Equation 2:
[tex]\[ 4x + 2y = 4(2) + 2(6) = 8 + 12 = 20 \quad \text{(satisfied)} \][/tex]
Both equations are satisfied, confirming our solution is correct.
#### Step 4: Graphing the System of Equations
While a detailed graph isn't shown here, you can plot both equations on a coordinate plane:
1. For [tex]\( x + y = 8 \)[/tex]:
- The line passes through points [tex]\( (8, 0) \)[/tex] and [tex]\( (0, 8) \)[/tex].
2. For [tex]\( 4x + 2y = 20 \)[/tex]:
- Divide by 2: [tex]\( 2x + y = 10 \)[/tex]
- The line passes through points [tex]\( (5, 0) \)[/tex] and [tex]\( (0, 10) \)[/tex].
The intersection point [tex]\( (2, 6) \)[/tex] represents the number of sheep and chickens, which matches our solution.
### Conclusion
The farmer has 2 sheep and 6 chickens.
Given the problem, we need to determine the number of sheep and chickens a farmer owns based on the total number of animals and legs.
#### Step 1: Set Up the System of Equations
Let's define our variables:
- [tex]\( x \)[/tex] = number of sheep
- [tex]\( y \)[/tex] = number of chickens
From the problem, we have the following two pieces of information:
1. The farmer has a total of 8 animals.
2. There is a total of 20 legs.
We can use this information to set up the following system of equations:
[tex]\[ \begin{cases} x + y = 8 \\ 4x + 2y = 20 \end{cases} \][/tex]
#### Step 2: Solve the System of Equations
Equation 1: [tex]\( x + y = 8 \)[/tex]
Equation 2: [tex]\( 4x + 2y = 20 \)[/tex]
Solve Equation 1 for [tex]\( y \)[/tex]:
[tex]\[ y = 8 - x \][/tex]
Substitute [tex]\( y = 8 - x \)[/tex] into Equation 2:
[tex]\[ 4x + 2(8 - x) = 20 \][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x + 16 - 2x = 20 \\ 2x + 16 = 20 \\ 2x = 4 \\ x = 2 \][/tex]
Now that we have [tex]\( x = 2 \)[/tex], substitute it back into the equation [tex]\( y = 8 - x \)[/tex]:
[tex]\[ y = 8 - 2 \\ y = 6 \][/tex]
Thus, the number of sheep is [tex]\( x = 2 \)[/tex] and the number of chickens is [tex]\( y = 6 \)[/tex].
#### Step 3: Verification
To ensure our solution is correct, we should verify it by plugging these values back into the original equations.
Verification with Equation 1:
[tex]\[ x + y = 2 + 6 = 8 \quad \text{(satisfied)} \][/tex]
Verification with Equation 2:
[tex]\[ 4x + 2y = 4(2) + 2(6) = 8 + 12 = 20 \quad \text{(satisfied)} \][/tex]
Both equations are satisfied, confirming our solution is correct.
#### Step 4: Graphing the System of Equations
While a detailed graph isn't shown here, you can plot both equations on a coordinate plane:
1. For [tex]\( x + y = 8 \)[/tex]:
- The line passes through points [tex]\( (8, 0) \)[/tex] and [tex]\( (0, 8) \)[/tex].
2. For [tex]\( 4x + 2y = 20 \)[/tex]:
- Divide by 2: [tex]\( 2x + y = 10 \)[/tex]
- The line passes through points [tex]\( (5, 0) \)[/tex] and [tex]\( (0, 10) \)[/tex].
The intersection point [tex]\( (2, 6) \)[/tex] represents the number of sheep and chickens, which matches our solution.
### Conclusion
The farmer has 2 sheep and 6 chickens.
