Express the problem as a system of equations using slack variables. (Use s1, s2, ... for the slack variables.)

Maximize
z = 150x1 + 240x2,
subject to
12x1	 + 	15x2	 ≤ 	50
8x1	 + 	22x2	 ≤ 	65
x1	 ≥ 	0, x2	 ≥ 	0.

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.

Express the problem as a system of equations using slack variables. (Use s1, s2, ... for the slack variables.) Maximize z = 150x1 + 240x2, subject to 12x1 + 15x2 ≤ 50, 8x1 + 22x2 ≤ 65, x1 ≥ 0, x2 ≥ 0.

Learn to express and solve a linear programming problem by converting inequalities into equations using slack variables. This problem involves maximizing the function z = 150x1 + 240x2, with constraints 12x1 + 15x2 ≤ 50 and 8x1 + 22x2 ≤ 65, using the simplex method. Suitable for advanced high school and undergraduate students studying optimization.

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