This question is about applying the Method of Lagrange Multipliers to find the maximum and minimum values of a function $f(x, y, z) = 2x + 3y - z$ subject to the constraint $(x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4$. The task is to determine how many of the given equations must be solved simultaneously.

Method of Lagrange Multipliers

Given a function $f(x, y, z)$ to optimize, subject to a constraint $g(x, y, z) = 0$, we introduce a Lagrange multiplier $\lambda$ and solve the system of equations:

$\nabla f = \lambda \nabla g$ and $g(x, y, z) = 0.$

Here, the constraint is: $g(x, y, z) = (x - 1)^2 + (y - 2)^2 + (z + 3)^2 - 4 = 0.$

System of Equations

From the given problem, the partial derivatives of $f(x, y, z)$ and $g(x, y, z)$ are:

1. $\frac{\partial f}{\partial x} = 2, \quad \frac{\partial g}{\partial x} = 2(x - 1)$,
2. $\frac{\partial f}{\partial y} = 3, \quad \frac{\partial g}{\partial y} = 2(y - 2)$,
3. $\frac{\partial f}{\partial z} = -1, \quad \frac{\partial g}{\partial z} = 2(z + 3)$.

Setting $\nabla f = \lambda \nabla g$, we get the system: $2 = \lambda \cdot 2(x - 1),$ $3 = \lambda \cdot 2(y - 2),$ $-1 = \lambda \cdot 2(z + 3).$ Plus the constraint equation: $(x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4.$

Given Equations

The equations in the question are: (i) $2x + 3y - z = 0$ (This is unrelated to the system derived from Lagrange multipliers), (ii) $2 = \lambda$, (iii) $3 = 2\lambda(y - 2)$, (iv) $1 = 2\lambda(z + 3)$, (v) $(x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4$.

To solve the optimization problem, we use equations (ii), (iii), (iv), and (v). Hence, 4 equations must be solved simultaneously.

The correct answer is 4.

Would you like further clarification on this? Here are some additional questions to consider:

1. How do Lagrange multipliers work in multiple dimensions?
2. Why is the constraint equation always included in the system?
3. How do you find partial derivatives for optimization?
4. What happens if there are multiple constraints?
5. How do you interpret the value of the Lagrange multiplier $\lambda$?

Tip: When using Lagrange multipliers, always check the constraint equation and all partial derivatives carefully for accuracy in applying the method.