`x + 2y le 40, 2x + y le 40, x ge 0, y ge 0`

Explanation

To solve the given system of linear inequalities step by step, we will follow these steps:

### Step 1: Write the inequalities
We are given the following inequalities:
1. \( x + 2y \leq 40 \)
2. \( 2x + y \leq 40 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert inequalities to equations
To find the boundary lines, we convert the inequalities into equations:
1. \( x + 2y = 40 \)
2. \( 2x + y = 40 \)

### Step 3: Find intercepts for the first equation
For the equation \( x + 2y = 40 \):
- Set \( x = 0 \):
\[ 0 + 2y = 40 \implies 2y = 40 \implies y = 20 \]
So, the y-intercept is \( (0, 20) \).

- Set \( y = 0 \):
\[ x + 0 = 40 \implies x = 40 \]
So, the x-intercept is \( (40, 0) \).

### Step 4: Find intercepts for the second equation
For the equation \( 2x + y = 40 \):
- Set \( x = 0 \):
\[ 2(0) + y = 40 \implies y = 40 \]
So, the y-intercept is \( (0, 40) \).

- Set \( y = 0 \):
\[ 2x + 0 = 40 \implies 2x = 40 \implies x = 20 \]
So, the x-intercept is \( (20, 0) \).

### Step 5: Plot the lines
Now we plot the points on a graph:
- For \( x + 2y = 40 \), plot the points \( (0, 20) \) and \( (40, 0) \).
- For \( 2x + y = 40 \), plot the points \( (0, 40) \) and \( (20, 0) \).

### Step 6: Determine the shaded region
Since both inequalities are less than or equal to, we will shade the region below each line:
- For \( x + 2y \leq 40 \), shade below the line connecting \( (0, 20) \) and \( (40, 0) \).
- For \( 2x + y \leq 40 \), shade below the line connecting \( (0, 40) \) and \( (20, 0) \).

### Step 7: Find the feasible region
The feasible region is where the shaded areas overlap, considering \( x \geq 0 \) and \( y \geq 0 \). This will be a polygon formed by the points:
- \( (0, 0) \)
- \( (0, 20) \)
- \( (20, 0) \)
- \( (0, 40) \)
- \( (40, 0) \)

### Step 8: Identify the vertices of the feasible region
The vertices of the feasible region can be found by identifying the intersection points of the lines:
1. The intersection of \( x + 2y = 40 \) and \( 2x + y = 40 \):
- Solve the equations simultaneously to find the intersection point.

### Conclusion
The solution to the system of inequalities is the set of all points \( (x, y) \) that lie within the shaded region defined by the inequalities.