`x + 2y le 10, x + y ge 1, x-y le 0, x ge 0, yge 0`

Explanation

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

### Step 1: Write the inequalities
The inequalities we need to solve are:
1. \( x + 2y \leq 10 \)
2. \( x + y \geq 1 \)
3. \( x - y \leq 0 \)
4. \( x \geq 0 \)
5. \( y \geq 0 \)

### Step 2: Convert inequalities to equations
To graph these inequalities, we first convert them into equations:
1. \( x + 2y = 10 \)
2. \( x + y = 1 \)
3. \( x - y = 0 \) (which simplifies to \( x = y \))

### Step 3: Find intercepts for each equation
- **For \( x + 2y = 10 \)**:
- When \( x = 0 \): \( 2y = 10 \) → \( y = 5 \) (y-intercept)
- When \( y = 0 \): \( x = 10 \) (x-intercept)

Intercepts: \( (10, 0) \) and \( (0, 5) \)

- **For \( x + y = 1 \)**:
- When \( x = 0 \): \( y = 1 \) (y-intercept)
- When \( y = 0 \): \( x = 1 \) (x-intercept)

Intercepts: \( (1, 0) \) and \( (0, 1) \)

- **For \( x - y = 0 \)** (or \( x = y \)):
- This line passes through the origin \( (0, 0) \) and has a slope of 1.

### Step 4: Plot the lines on a graph
Now, we will plot the lines based on the intercepts found:
1. Draw the line for \( x + 2y = 10 \) using points \( (10, 0) \) and \( (0, 5) \).
2. Draw the line for \( x + y = 1 \) using points \( (1, 0) \) and \( (0, 1) \).
3. Draw the line for \( x = y \) which is a diagonal line through the origin.

### Step 5: Determine the shaded regions
- **For \( x + 2y \leq 10 \)**: Test the point \( (0, 0) \):
- \( 0 + 2(0) \leq 10 \) → True, so shade towards the origin.

- **For \( x + y \geq 1 \)**: Test the point \( (0, 0) \):
- \( 0 + 0 \geq 1 \) → False, so shade away from the origin.

- **For \( x - y \leq 0 \)**: Test the point \( (0, 0) \):
- \( 0 - 0 \leq 0 \) → True, so shade above the line \( x = y \).

### Step 6: Identify the feasible region
The feasible region is the area where all shaded regions overlap, considering the constraints \( x \geq 0 \) and \( y \geq 0 \).

### Step 7: Conclusion
The common shaded area represents the solution to the system of inequalities.