Answer :
### Step-by-Step Solution
Let's address each part of the problem in detail:
#### 1) Writing the System of Equations and Graphing It
You are given two types of animals: sheep and chickens. The problem states:
- Sheep have 4 legs each.
- Chickens have 2 legs each.
- The total number of animals is 8.
- The total number of legs is 20.
We'll define:
- [tex]\( x \)[/tex]: the number of sheep.
- [tex]\( y \)[/tex]: the number of chickens.
This situation can be modeled using the following system of linear equations:
1. The total number of animals:
[tex]\[ x + y = 8 \][/tex]
2. The total number of legs:
[tex]\[ 4x + 2y = 20 \][/tex]
Next, let's look at the graphical representation. These equations represent two lines on a coordinate plane where the x-axis represents the number of sheep and the y-axis represents the number of chickens.
To graph these equations:
- For [tex]\( x + y = 8 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 8 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 8 \)[/tex]
- Plot the points (0,8) and (8,0) and draw the line through them.
- For [tex]\( 4x + 2y = 20 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( 2y = 20 \)[/tex] → [tex]\( y = 10 \)[/tex] (but this point won't be used since [tex]\( x + y = 8 \)[/tex])
- When [tex]\( y = 0 \)[/tex]: [tex]\( 4x = 20 \)[/tex] → [tex]\( x = 5 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 4 \cdot 2 + 2y = 20 \)[/tex] → [tex]\( 2y = 20 - 8 \)[/tex] → [tex]\( 2y = 12 \)[/tex] → [tex]\( y = 6 \)[/tex]
- Plot points (0,10), (5,0), and (2,6) to draw the line (trimming the graph at [tex]\( x=0 \)[/tex] and [tex]\( x=5 \)[/tex] since they must sum up [tex]\( x + y = 8 \)[/tex]).
These two lines will intersect, and their intersection point represents the solution to the system of equations.
#### 2) Determining the Number of Sheep and Chickens
Using the system of equations:
[tex]\[ x + y = 8 \][/tex]
[tex]\[ 4x + 2y = 20 \][/tex]
To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
1. From the first equation, solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 8 - x \][/tex]
2. Substitute [tex]\( y = 8 - x \)[/tex] into the second equation:
[tex]\[ 4x + 2(8 - x) = 20 \][/tex]
This simplifies to:
[tex]\[ 4x + 16 - 2x = 20 \][/tex]
[tex]\[ 2x + 16 = 20 \][/tex]
[tex]\[ 2x = 4 \][/tex]
[tex]\[ x = 2 \][/tex]
3. Substitute [tex]\( x = 2 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ 2 + y = 8 \][/tex]
[tex]\[ y = 6 \][/tex]
Therefore, the number of sheep is [tex]\( x = 2 \)[/tex] and the number of chickens is [tex]\( y = 6 \)[/tex].
### Conclusion
The farmer has 2 sheep and 6 chickens.
Let's address each part of the problem in detail:
#### 1) Writing the System of Equations and Graphing It
You are given two types of animals: sheep and chickens. The problem states:
- Sheep have 4 legs each.
- Chickens have 2 legs each.
- The total number of animals is 8.
- The total number of legs is 20.
We'll define:
- [tex]\( x \)[/tex]: the number of sheep.
- [tex]\( y \)[/tex]: the number of chickens.
This situation can be modeled using the following system of linear equations:
1. The total number of animals:
[tex]\[ x + y = 8 \][/tex]
2. The total number of legs:
[tex]\[ 4x + 2y = 20 \][/tex]
Next, let's look at the graphical representation. These equations represent two lines on a coordinate plane where the x-axis represents the number of sheep and the y-axis represents the number of chickens.
To graph these equations:
- For [tex]\( x + y = 8 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 8 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 8 \)[/tex]
- Plot the points (0,8) and (8,0) and draw the line through them.
- For [tex]\( 4x + 2y = 20 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( 2y = 20 \)[/tex] → [tex]\( y = 10 \)[/tex] (but this point won't be used since [tex]\( x + y = 8 \)[/tex])
- When [tex]\( y = 0 \)[/tex]: [tex]\( 4x = 20 \)[/tex] → [tex]\( x = 5 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 4 \cdot 2 + 2y = 20 \)[/tex] → [tex]\( 2y = 20 - 8 \)[/tex] → [tex]\( 2y = 12 \)[/tex] → [tex]\( y = 6 \)[/tex]
- Plot points (0,10), (5,0), and (2,6) to draw the line (trimming the graph at [tex]\( x=0 \)[/tex] and [tex]\( x=5 \)[/tex] since they must sum up [tex]\( x + y = 8 \)[/tex]).
These two lines will intersect, and their intersection point represents the solution to the system of equations.
#### 2) Determining the Number of Sheep and Chickens
Using the system of equations:
[tex]\[ x + y = 8 \][/tex]
[tex]\[ 4x + 2y = 20 \][/tex]
To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
1. From the first equation, solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 8 - x \][/tex]
2. Substitute [tex]\( y = 8 - x \)[/tex] into the second equation:
[tex]\[ 4x + 2(8 - x) = 20 \][/tex]
This simplifies to:
[tex]\[ 4x + 16 - 2x = 20 \][/tex]
[tex]\[ 2x + 16 = 20 \][/tex]
[tex]\[ 2x = 4 \][/tex]
[tex]\[ x = 2 \][/tex]
3. Substitute [tex]\( x = 2 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ 2 + y = 8 \][/tex]
[tex]\[ y = 6 \][/tex]
Therefore, the number of sheep is [tex]\( x = 2 \)[/tex] and the number of chickens is [tex]\( y = 6 \)[/tex].
### Conclusion
The farmer has 2 sheep and 6 chickens.
