To address the problem at hand, we need to translate the context into mathematical constraints. Let's break down the requirements and constraints step-by-step:
1. Total Bags Requirement: The farm must order at least 60 bags per week.
- This translates to: [tex]\( x + y \geq 60 \)[/tex].
2. Store Supplier Constraint: The farm commits to purchasing at least as many bags from store [tex]\( X \)[/tex] as from store [tex]\( Y \)[/tex].
- This translates to: [tex]\( x \geq y \)[/tex].
3. Maximum Bags from Store [tex]\( Y \)[/tex]: Store [tex]\( Y \)[/tex] can supply a maximum of 40 bags per week.
- This translates to: [tex]\( y \leq 40 \)[/tex].
4. Non-Negative Orders: The farm cannot order a negative number of bags from either store.
- This translates to: [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
Given these insights, let's match them to the options provided:
- Option a:
```
20x + 15y \geq 60
x \geq y
y \leq 40
x \geq 0
y \geq 0
```
The first constraint here [tex]\( 20x + 15y \geq 60 \)[/tex] does not correctly represent the total number of bags needed. This is instead expected to relate costs, not quantity.
- Option b:
```
x + y \geq 60
x \geq y
y \leq 40
x \geq 0
y \geq 0
```
This matches our identified constraints perfectly.
- Option c:
```
x + y \geq 60
y \geq x
y \leq 40
x \geq 0
y \geq 0
```
The second constraint [tex]\( y \geq x \)[/tex] is incorrect as it implies that the farm should order more or equal bags from the store [tex]\( Y \)[/tex] compared to the store [tex]\( X \)[/tex], which contradicts the problem statement.
- Option d:
```
x + y \leq 60
x \geq y
y \leq 40
x \geq 0
y \geq 0
```
The first constraint [tex]\( x + y \leq 60 \)[/tex] is incorrect because it contradicts the requirement that the farm must order at least 60 bags per week.
Therefore, the correct set of constraints is in option b:
[tex]\[
\begin{array}{l}
x + y \geq 60 \\
x \geq y \\
y \leq 40 \\
x \geq 0 \\
y \geq 0
\end{array}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
