https://file.aiquickdraw.com/mathbot/file/1725423495767remg1vc5.jpg

Resolve corretamentr cada uma das questoes por favor Pregunta 11
Resolver el sistema de ecuaciones −x + y + 2z = 1, 2x + 3y + z = −2, 5x + 4y + 2z = 4
x = 2, y = −3, z = 0x = 2, y = 3, z = −3x = 2, y = −1, z = 3x = 2, y = −3, z = 3x = 1, y = −3, z = 3

It appears that you've uploaded an image containing a graph, which is likely related to a pathfinding problem, and you've also included a system of equations in your text that you want to solve. I'll address each part step by step.

### **Part 1: Graph-Related Problem**
Given the image, it looks like a typical network flow or critical path method (CPM) graph where nodes represent events and edges represent activities with associated weights or durations. However, without specific questions or instructions related to the graph, I'll focus on solving the system of equations provided.

### **Part 2: Solving the System of Equations**

The system of equations provided is:

\[
\begin{aligned}
- x + y + 2z &= 1 \quad \text{(Equation 1)} \\
2x + 3y + z &= -2 \quad \text{(Equation 2)} \\
5x + 4y + 2z &= 4 \quad \text{(Equation 3)}
\end{aligned}
\]

Let's solve this system using the method of substitution or elimination.

### **Step 1: Solve for One Variable**

We'll use the elimination method to simplify the equations. First, let's multiply Equation 1 by 2 so that we can eliminate \( z \) by subtracting Equation 2 from the modified Equation 1:

\[
\text{Modified Equation 1: } -2x + 2y + 4z = 2
\]
\[
\text{Equation 2: } 2x + 3y + z = -2
\]

Now, subtract Equation 2 from the modified Equation 1:

\[
(-2x + 2y + 4z) - (2x + 3y + z) = 2 - (-2)
\]
\[
-4x - y + 3z = 4 \quad \text{(Equation 4)}
\]

### **Step 2: Eliminate Another Variable**

Now, let's subtract Equation 1 directly from Equation 3 to further eliminate variables:

\[
\text{Equation 3: } 5x + 4y + 2z = 4
\]
\[
\text{Equation 1: } -x + y + 2z = 1
\]
\[
(5x + 4y + 2z) - (-x + y + 2z) = 4 - 1
\]
\[
6x + 3y = 3 \quad \text{(Equation 5)}
\]

### **Step 3: Solve the Reduced System**

Now we have two equations:

\[
\begin{aligned}
-4x - y + 3z &= 4 \quad \text{(Equation 4)} \\
6x + 3y &= 3 \quad \text{(Equation 5)}
\end{aligned}
\]

From Equation 5, solve for \( y \):

\[
y = \frac{3 - 6x}{3} = 1 - 2x
\]

Substitute \( y = 1 - 2x \) into Equation 4:

\[
-4x - (1 - 2x) + 3z = 4
\]
\[
-4x - 1 + 2x + 3z = 4
\]
\[
-2x + 3z = 5 \quad \text{(Equation 6)}
\]

### **Step 4: Solve for \( x \), \( y \), and \( z \)**

Let's solve Equation 6 for \( z \):

\[
z = \frac{5 + 2x}{3}
\]

Substitute this value of \( z \) back into the original Equation 1:

\[
-x + (1 - 2x) + 2\left(\frac{5 + 2x}{3}\right) = 1
\]

Simplify and solve for \( x \):

\[
-x - 2x + 1 + \frac{10 + 4x}{3} = 1
\]
Multiply everything by 3 to eliminate the fraction:

\[
-9x + 3 + 10 + 4x = 3
\]
\[
-5x + 13 = 3
\]
\[
-5x = -10 \quad \Rightarrow \quad x = 2
\]

Now, use \( x = 2 \) to find \( y \) and \( z \):

\[
y = 1 - 2(2) = 1 - 4 = -3
\]
\[
z = \frac{5 + 4}{3} = 3
\]

### **Final Answer:**
The solution is \( x = 2 \), \( y = -3 \), \( z = 3 \). Therefore, the correct answer among the options provided is:

**\( x = 2, y = -3, z = 3 \)**

---

Would you like further details on how the calculations were done, or do you have any specific questions?

### **Related Questions:**
1. How can you solve a system of equations using matrix methods?
2. What are other methods to solve systems of linear equations?
3. How do you interpret the solutions of a system of linear equations graphically?
4. What is the importance of checking solutions in a system of equations?
5. How can you apply these methods to non-linear systems?

### **Tip:** When solving systems of equations, always double-check your substitutions and simplifications to ensure accuracy.

Resolve corretamente cada uma das questoes por favor. Pregunta 11 - Resolver el sistema de ecuaciones −x + y + 2z = 1, 2x + 3y + z = −2, 5x + 4y + 2z = 4

This page provides a step-by-step solution to solving a system of linear equations using substitution and elimination methods. The problem involves three variables (x, y, z) and is suitable for students in Grades 9-12.

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