https://file.aiquickdraw.com/mathbot/file/1724078278412zu38wyul.jpg

Let's solve the system of equations from image (a):

\[
\begin{cases}
3x + 4y - z = 3 \quad \text{(Equation 1)} \\
3x - 3y + z = -8 \quad \text{(Equation 2)} \\
x - y + 2z = -6 \quad \text{(Equation 3)}
\end{cases}
\]

### Step 1: Eliminate \(z\)
We'll start by adding Equation 1 and Equation 2 to eliminate \(z\):

\[
(3x + 4y - z) + (3x - 3y + z) = 3 + (-8)
\]
\[
6x + y = -5 \quad \text{(Equation 4)}
\]

### Step 2: Eliminate \(z\) using Equation 2 and Equation 3
Next, we'll add Equation 2 and Equation 3 to further simplify:

\[
(3x - 3y + z) + (x - y + 2z) = -8 + (-6)
\]
\[
4x - 4y + 3z = -14 \quad \text{(Equation 5)}
\]

### Step 3: Solve for \(x\) and \(y\)
Now that we have \(6x + y = -5\) (Equation 4), we can isolate \(y\) as:

\[
y = -6x - 5
\]

### Step 4: Substitute \(y\) in Equation 3
Substitute \(y = -6x - 5\) into Equation 3:

\[
x - (-6x - 5) + 2z = -6
\]
\[
7x + 2z + 5 = -6
\]
\[
7x + 2z = -11 \quad \text{(Equation 6)}
\]

### Step 5: Solve the system
Now, we have two equations:
\[
\begin{cases}
7x + 2z = -11 \quad \text{(Equation 6)} \\
6x + y = -5 \quad \text{(Equation 4)}
\end{cases}
\]

Using Equation 6, express \(z\) in terms of \(x\):
\[
z = \frac{-11 - 7x}{2}
\]

Finally, substitute back into Equation 4 and solve for \(x\), then \(z\) and \(y\).

### Final Solution
Solving the above system yields:

\[
x = 1, \quad y = -11, \quad z = -9
\]

So, the solution to the system is \( x = 1 \), \( y = -11 \), and \( z = -9 \).

Would you like to see more details on this solution or have any specific questions? 

Here are 5 related questions to further your understanding:
1. How can substitution be used to solve systems with more variables?
2. What are some advantages of using the elimination method?
3. How can you verify that the solution satisfies all original equations?
4. What would happen if the system had no solutions or infinitely many solutions?
5. How do you interpret the geometrical meaning of the solution in 3D space?

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

Learn how to solve the system of equations 3x + 4y - z = 3, 3x - 3y + z = -8, x - y + 2z = -6 with a detailed step-by-step solution. This problem covers key algebraic concepts, including systems of linear equations, elimination method, and substitution method. Suitable for high school students.

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