The given matrix represents a simplex tableau, which is used to solve linear programming problems. To interpret the tableau and find the solution, we need to analyze the basic variables and non-basic variables in the current tableau.

Step-by-step approach to interpret the solution:

• First row: $x_1, x_2, x_3, s_1, s_2, s_3, z$ and the right-hand side (RHS).

From the tableau:

x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & z & RHS \\ \hline 2 & 0 & 10 & 2 & 0 & 0 & 0 & 50 \\ 0 & 3 & 0 & 5 & 0 & 0 & 0 & 15 \\ 0 & 0 & 0 & 0 & 2 & 22 & 0 & 8 \\ -3 & 0 & 0 & 8 & 0 & 8 & 2 & 26 \\ \end{array}$$ - **Basic Variables**: These are the variables corresponding to the columns with a unit matrix (1 in a single row and 0s elsewhere). From the tableau, we can see that $$s_1$$, $$s_2$$, and $$s_3$$ are the basic variables, as they correspond to the rows with 1's in their respective columns. - **Non-basic Variables**: These are the variables corresponding to the columns that don't form the unit matrix. In this case, $$x_1$$, $$x_2$$, and $$x_3$$ are non-basic variables because they don't have 1's in their columns. ### Basic Variables and Their Values: - $$s_1$$ (Slack variable for the first constraint): From the first row, $$s_1 = 50$$. - $$s_2$$ (Slack variable for the second constraint): From the second row, $$s_2 = 15$$. - $$s_3$$ (Slack variable for the third constraint): From the third row, $$s_3 = 8$$. ### Non-basic Variables: - Since the non-basic variables are not currently in the solution (because they are not in the basic columns), their values will be **0**. - Therefore: - $$x_1 = 0$$ - $$x_2 = 0$$ - $$x_3 = 0$$ ### Optimal Objective Function Value: From the last row, the value of the objective function (z) is given by **26**, so the **optimal value of z** is **26**. ### Solution: The current solution to the system is: - $$x_1 = 0$$ - $$x_2 = 0$$ - $$x_3 = 0$$ - $$s_1 = 50$$ - $$s_2 = 15$$ - $$s_3 = 8$$ - **Objective value (z) = 26** Would you like any further details on the interpretation of the tableau or the solution process? --- Here are five related questions for deeper understanding: 1. What does the value of $$z$$ represent in the context of linear programming? 2. How do you identify the basic and non-basic variables in a simplex tableau? 3. What would happen if a non-basic variable had a positive value in the tableau? 4. How do you perform the pivot operation in the simplex method to move to the next tableau? 5. What is the significance of slack variables in a linear programming problem? **Tip**: In linear programming, the objective function's optimal value can sometimes be found without fully solving the problem if all the entries in the last row (except for the RHS) are non-positive.