To solve this linear programming problem using the simplex method, follow these steps:

Problem Setup:

Objective Function: $P = -x_1 + 2x_2$

Constraints:

1. $-x_1 + x_2 \leq 2$
2. $-x_1 + 3x_2 \leq 14$
3. $x_1 + 4x_2 \leq 10$
4. $x_1, x_2 \geq 0$

Step-by-Step Solution Using the Simplex Method:

1. Convert inequalities to equalities by adding slack variables. Let $s_1, s_2, s_3 \geq 0$ be slack variables.

   • $-x_1 + x_2 + s_1 = 2$
   • $-x_1 + 3x_2 + s_2 = 14$
   • $x_1 + 4x_2 + s_3 = 10$

2. Initial Simplex Tableau:

3. Pivot and Iterate:

   • Identify the pivot column (most negative coefficient in the objective row, indicating increase in $P$).
   • Calculate pivot row using the minimum ratio test.
   • Perform row operations to make the pivot element 1 and other elements in the column 0.
   • Update tableau and repeat until all entries in the objective row are non-negative.

4. Find Solution:

   • After final tableau, determine the values of $x_1$, $x_2$, and slack variables.
   • Compute $P = -x_1 + 2x_2$.

Would you like me to proceed with the detailed simplex iterations for this problem?