To express this problem as a system of equations using slack variables, we'll introduce slack variables to convert the inequalities into equalities. Let's proceed with each constraint.

Original Problem

Maximize: $z = 150x_1 + 240x_2$

Subject to:

1. $12x_1 + 15x_2 \leq 50$
2. $8x_1 + 22x_2 \leq 65$
3. $x_1 \geq 0, x_2 \geq 0$

Step 1: Introduce Slack Variables

For each inequality constraint, we add a slack variable to turn the inequalities into equalities. Let $s_1$ and $s_2$ be the slack variables for the constraints.

1. For $12x_1 + 15x_2 \leq 50$, we add slack variable $s_1$: $12x_1 + 15x_2 + s_1 = 50$

2. For $8x_1 + 22x_2 \leq 65$, we add slack variable $s_2$: $8x_1 + 22x_2 + s_2 = 65$

Step 2: Write the System of Equations

Now, we have the following system of equations with the slack variables included:

Maximize: $z = 150x_1 + 240x_2$

Subject to: $12x_1 + 15x_2 + s_1 = 50$ $8x_1 + 22x_2 + s_2 = 65$ $x_1 \geq 0, \; x_2 \geq 0, \; s_1 \geq 0, \; s_2 \geq 0$

This formulation is now ready for application in the simplex method or other linear programming techniques.

Would you like more details on the next steps for solving this system, or do you have any questions?

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Questions for Further Exploration:

1. What are slack variables and why are they used in linear programming?
2. How does the simplex method utilize this system to find the maximum value of $z$?
3. How would the system change if there was an additional constraint?
4. What are feasible and optimal solutions in linear programming?
5. How can we interpret the economic meaning of $s_1$ and $s_2$ once we find their values?

Tip: When adding slack variables, ensure each inequality is converted to equality before using linear programming methods like the simplex method.