To solve this problem using the simplex method, we need to maximize the objective function:

$P = 8x_1 + 15x_2$

subject to the constraints:

$5x_1 + x_2 \leq 10$ $x_1 + 5x_2 \leq 15$ $x_1 \geq 0, \quad x_2 \geq 0$

Step 1: Convert inequalities to equations

To apply the simplex method, we need to rewrite the inequalities as equations by introducing slack variables.

1. For $5x_1 + x_2 \leq 10$, introduce slack variable $s_1$: $5x_1 + x_2 + s_1 = 10$

2. For $x_1 + 5x_2 \leq 15$, introduce slack variable $s_2$: $x_1 + 5x_2 + s_2 = 15$

Now the system becomes:

$P = 8x_1 + 15x_2$ subject to: $5x_1 + x_2 + s_1 = 10$ $x_1 + 5x_2 + s_2 = 15$ $x_1, x_2, s_1, s_2 \geq 0$

Step 2: Set up the initial simplex tableau

The tableau for the initial system is: