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The problem involves using the Method of Lagrange Multipliers to find the maximum and minimum values of the function \( f(x, y, z) = 2x + 3y - z \) under the constraint \( (x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4 \). We need to determine how many of the given equations are part of the system that needs to be solved simultaneously.

The equations listed are:
1. \( 2x + 3y - z = 0 \)
2. \( 2 = \lambda \cdot 2(x - 1) \)
3. \( 3 = \lambda \cdot 2(y - 2) \)
4. \( 1 = \lambda \cdot 2(z + 3) \)
5. \( (x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4 \)

### Analyzing the Equations for the Lagrange Multipliers Method
The Method of Lagrange Multipliers requires solving the following system:
- The gradient of the function \( f(x, y, z) \) set equal to a scalar multiple (\( \lambda \)) of the gradient of the constraint function.
- The constraint equation itself.

For the given function \( f(x, y, z) = 2x + 3y - z \) and the constraint \( g(x, y, z) = (x - 1)^2 + (y - 2)^2 + (z + 3)^2 - 4 = 0 \), the equations are:

1. Gradient equation for \( x \): \( \frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x} \), which is \( 2 = \lambda \cdot 2(x - 1) \).
2. Gradient equation for \( y \): \( \frac{\partial f}{\partial y} = \lambda \frac{\partial g}{\partial y} \), which is \( 3 = \lambda \cdot 2(y - 2) \).
3. Gradient equation for \( z \): \( \frac{\partial f}{\partial z} = \lambda \frac{\partial g}{\partial z} \), which is \( -1 = \lambda \cdot 2(z + 3) \).
4. The constraint itself: \( (x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4 \).

### Conclusion
Based on the analysis, the correct set of equations includes:
- Equation (ii): \( 2 = \lambda \cdot 2(x - 1) \)
- Equation (iii): \( 3 = \lambda \cdot 2(y - 2) \)
- Equation (iv): \( 1 = \lambda \cdot 2(z + 3) \)
- Equation (v): \( (x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4 \)

So, there are **4 equations** in the system that must be solved simultaneously.

### Answer
The answer is **4**.

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

Here are five follow-up questions related to the topic:

1. What is the purpose of the Lagrange multiplier (\( \lambda \)) in the method?
2. How do you find the gradients of the functions involved in Lagrange Multipliers?
3. What are some common applications of the Lagrange Multipliers method in real-world problems?
4. How would you approach solving the system of equations obtained through this method?
5. What happens if multiple constraints are involved in the Lagrange Multipliers method?

**Tip:** When solving using Lagrange Multipliers, always verify that the constraint equation is satisfied after finding the critical points.

When using the Method of Lagrange Multipliers to find the maximum and minimum values of f(x, y, z) = 2x + 3y - z subject to the constraint (x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4, how many of the following equations are included in the system which must be simultaneously solved?

Use the Lagrange Multipliers method to find the maximum and minimum values of f(x, y, z) = 2x + 3y - z subject to the constraint (x - 1)^2 + (y - 2)^2 + (z + 3)^2 = 4. This detailed step-by-step solution demonstrates solving four simultaneous equations, covering concepts in multivariable calculus.

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