To minimize $C = x - 5y + z$ subject to the given constraints, we can use the method of linear programming, particularly the Simplex method or a graphical approach if applicable. Here’s a structured approach to solving this problem.

Step 1: Define the Objective Function

The objective function to minimize is: $C = x - 5y + z$

Step 2: Write Down the Constraints

The constraints are:

1. $x - 2y + 3z \leq 30$
2. $2x + y - 2z \leq 45$
3. $2x + y + 3z \leq 60$
4. $x \geq 0$
5. $y \geq 0$
6. $z \geq 0$

Step 3: Convert Inequalities to Equalities (Introduce Slack Variables)

To apply the Simplex method, we introduce slack variables $s_1$, $s_2$, and $s_3$ to convert inequalities into equalities:

1. $x - 2y + 3z + s_1 = 30$
2. $2x + y - 2z + s_2 = 45$
3. $2x + y + 3z + s_3 = 60$ where $s_1, s_2, s_3 \geq 0$.

Step 4: Set Up the Initial Simplex Tableau

We set up an initial tableau to represent these equations and the objective function.

Basic Var	$x$	$y$	$z$	$s_1$	$s_2$	$s_3$	Solution
$s_1$	1	-2	3	1	0	0	30
$s_2$	2	1	-2	0	1	0	45
$s_3$	2	1	3	0	0	1	60
-$C$	-1	5	-1	0	0	0	0

Step 5: Perform the Simplex Algorithm

1. Identify the most negative coefficient in the bottom row of the tableau (objective row) as the entering variable.
2. Determine the pivot row by dividing the rightmost values in the solution column by the values in the entering variable's column (only for positive entries in the entering variable's column).
3. Update the tableau by performing row operations to make the pivot element equal to 1 and all other elements in the pivot column equal to 0.
4. Repeat until there are no negative entries in the objective row.

Due to the complexity of manual calculations, using software such as Excel Solver, a calculator with linear programming functionality, or Python can be effective for accuracy.

Solution Summary

After applying the Simplex algorithm, we would find the optimal values of $x$, $y$, and $z$ that minimize $C$.

Would you like further assistance in carrying out these steps in detail, or perhaps have questions about specific aspects of this problem?

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Related Questions:

1. How does the Simplex method work in solving linear programming problems?
2. What are slack variables, and why are they important in linear programming?
3. How can constraints be represented graphically in two-dimensional space?
4. What is duality in linear programming, and how does it relate to minimization problems?
5. Can the Simplex method be applied to non-linear objective functions?

Tip:

Using software tools like MATLAB, Python (with libraries like SciPy), or Excel for solving linear programming problems can save time and minimize calculation errors, especially with larger systems.