https://file.aiquickdraw.com/mathbot/file/1729180169028xbkzp9mz.jpg

The system of equations shown in the image is:

\[
\begin{aligned}
1. & \quad 3x + 2y - z = 8 \\
2. & \quad 2x + 2z = -4 \\
3. & \quad x + 3y = 4
\end{aligned}
\]

We will solve this system step by step using substitution or elimination methods to find the values of \(x\), \(y\), and \(z\).

### Step 1: Solve Equation (3) for \(x\)
From the third equation:
\[
x + 3y = 4 \quad \Rightarrow \quad x = 4 - 3y
\]

### Step 2: Substitute \(x = 4 - 3y\) into Equations (1) and (2)
#### For Equation (1):
Substitute \(x = 4 - 3y\) into \(3x + 2y - z = 8\):
\[
3(4 - 3y) + 2y - z = 8
\]
Simplifying:
\[
12 - 9y + 2y - z = 8 \quad \Rightarrow \quad 12 - 7y - z = 8
\]
\[
-7y - z = -4 \quad \Rightarrow \quad z = -7y + 4 \quad \text{(Equation A)}
\]

#### For Equation (2):
Substitute \(x = 4 - 3y\) into \(2x + 2z = -4\):
\[
2(4 - 3y) + 2z = -4
\]
Simplifying:
\[
8 - 6y + 2z = -4 \quad \Rightarrow \quad 2z = -4 + 6y - 8
\]
\[
2z = 6y - 12 \quad \Rightarrow \quad z = 3y - 6 \quad \text{(Equation B)}
\]

### Step 3: Set Equations A and B Equal to Each Other
From Equations A and B:
\[
-7y + 4 = 3y - 6
\]
Solving for \(y\):
\[
-7y - 3y = -6 - 4 \quad \Rightarrow \quad -10y = -10 \quad \Rightarrow \quad y = 1
\]

### Step 4: Substitute \(y = 1\) into the Expressions for \(x\) and \(z\)
- From \(x = 4 - 3y\):
\[
x = 4 - 3(1) = 1
\]
- From \(z = 3y - 6\):
\[
z = 3(1) - 6 = -3
\]

### Final Solution:
\[
x = 1, \quad y = 1, \quad z = -3
\]

These are the values for \(x\), \(y\), and \(z\).

Do you need further details or clarifications on this solution?

Here are 5 related questions you might want to explore:
1. What is the general method for solving a system of three linear equations?
2. How does substitution differ from the elimination method?
3. How would the solution change if one of the equations were different?
4. Can a system of linear equations have no solution? Under what conditions?
5. How can matrix methods (like Gaussian elimination) be applied to this problem?

**Tip**: Always check your solution by substituting the values back into the original equations to ensure they satisfy all the equations.

Solve the following system of linear equations:
3x + 2y - z = 8
2x + 2z = -4
x + 3y = 4

Step-by-step solution for the system of linear equations 3x + 2y - z = 8, 2x + 2z = -4, and x + 3y = 4. Learn how to solve using substitution and elimination methods, with a clear breakdown of each step. Ideal for high school students (Grades 9-12).

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