`Z= 4x + 7y`
`x + 2y le 20, x + y le 15, x ge 0, y ge 0`.

Answer

To solve the linear programming problem given by the objective function \( Z = 4x + 7y \) with the constraints:

1. \( x + 2y \leq 20 \)
2. \( x + y \leq 15 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

we will follow these steps:

### Step 1: Identify the Constraints
The constraints are:
- \( x + 2y \leq 20 \)
- \( x + y \leq 15 \)
- \( x \geq 0 \)
- \( y \geq 0 \)