This problem involves minimizing the objective function $C = x - 3y + z$, subject to a set of linear inequalities. To solve it, we typically use the Simplex method or Graphical method (if it’s in two dimensions). Since this involves three variables ($x$, $y$, and $z$), the Simplex method or similar linear programming techniques will be applied.

Constraints:

1. $x - 2y + 3z \leq 10$
2. $2x + y - 2z \leq 15$
3. $2x + y + 3z \leq 20$
4. $x \geq 0, y \geq 0, z \geq 0$ (non-negativity conditions)

The general approach to solving this problem involves:

1. Forming the system of linear equations from the inequalities (which can be solved as equalities to find potential corner points).
2. Checking the values of the objective function $C = x - 3y + z$ at the corner points.
3. The corner point that gives the minimum value will be the solution.

Let me compute the corner points and evaluate the function at those points to find the minimum.The minimum value of the objective function $C = x - 3y + z$ is:

$C = -50$

This occurs at the point $(x, y, z) = (0, 17, 1)$.

Would you like more details or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. What is the Simplex method, and how does it work for solving linear programming problems?
2. Why do we convert inequalities into equalities by adding slack variables in linear programming?
3. How do we interpret corner points in the context of optimization problems?
4. What role does duality play in linear programming problems?
5. How can linear programming be applied to real-world resource allocation problems?

Tip: When solving optimization problems, always check for corner points or vertices, as the optimum value for linear programming problems will occur at one of them.