https://file.aiquickdraw.com/mathbot/file/1726429125506x7vcn2kh.jpg

The system of equations shown in the image is:

\[
\begin{aligned}
1. &\ 2x + y - z = -2 \\
2. &\ 3x + 6y - 8z = 2 \\
3. &\ 9x + 8y - 8z = -2
\end{aligned}
\]

We can solve this system of equations using a method such as substitution, elimination, or matrix methods. Let's proceed with the elimination method to find the values of \(x\), \(y\), and \(z\).

### Step 1: Eliminate one variable
First, let's eliminate \(z\) from equations (2) and (3) by subtracting them:

\[
(9x + 8y - 8z) - (3x + 6y - 8z) = (-2) - 2
\]

Simplifying:

\[
6x + 2y = -4
\]

So we now have:

\[
6x + 2y = -4 \tag{4}
\]

### Step 2: Eliminate \(z\) from another set of equations
Next, let's eliminate \(z\) from equations (1) and (2). To do this, we'll multiply equation (1) by 8 and add it to equation (2):

\[
8(2x + y - z) = 8(-2) \implies 16x + 8y - 8z = -16
\]

Now, add this to equation (2):

\[
(16x + 8y - 8z) + (3x + 6y - 8z) = -16 + 2
\]

Simplifying:

\[
19x + 14y = -14 \tag{5}
\]

### Step 3: Solve the system of two equations
Now, we solve the system of two equations we obtained:

\[
\begin{aligned}
6x + 2y &= -4 \\
19x + 14y &= -14
\end{aligned}
\]

We can multiply the first equation by 7 to align the coefficients of \(y\):

\[
7(6x + 2y) = 7(-4) \implies 42x + 14y = -28
\]

Now subtract this from equation (5):

\[
(19x + 14y) - (42x + 14y) = -14 - (-28)
\]

Simplifying:

\[
-23x = 14 \implies x = -\frac{14}{23}
\]

### Step 4: Substitute \(x\) back to find \(y\)
Substitute \(x = -\frac{14}{23}\) into equation (4):

\[
6\left(-\frac{14}{23}\right) + 2y = -4
\]

Simplifying:

\[
-\frac{84}{23} + 2y = -4
\]

Multiply through by 23 to eliminate the fraction:

\[
-84 + 46y = -92
\]

Solving for \(y\):

\[
46y = -8 \implies y = -\frac{4}{23}
\]

### Step 5: Solve for \(z\)
Substitute \(x = -\frac{14}{23}\) and \(y = -\frac{4}{23}\) into equation (1):

\[
2\left(-\frac{14}{23}\right) + \left(-\frac{4}{23}\right) - z = -2
\]

Simplifying:

\[
-\frac{28}{23} - \frac{4}{23} - z = -2
\]

\[
-\frac{32}{23} - z = -2
\]

Multiply through by 23:

\[
-32 - 23z = -46
\]

Solving for \(z\):

\[
23z = 14 \implies z = \frac{14}{23}
\]

### Final Answer:
The solution to the system is:

\[
x = -\frac{14}{23}, \quad y = -\frac{4}{23}, \quad z = \frac{14}{23}
\]

Would you like more details or have any further questions?

### Related Questions:
1. How can the substitution method be applied to solve this system of equations?
2. What are some other methods to solve systems of equations?
3. How can matrix operations be used to solve a system of equations?
4. What are some common pitfalls when solving systems of equations?
5. How can we verify the solution for this system of equations?

### Tip:
Always check your solution by substituting the values of \(x\), \(y\), and \(z\) back into the original equations to ensure they satisfy all equations.

Solve the following system of equations for all three variables:
2x + y - z = -2
3x + 6y - 8z = 2
9x + 8y - 8z = -2

Learn how to solve a system of linear equations with three variables, using methods like elimination and substitution. This example covers algebraic methods, simplifying equations, and solving for unknowns. Suitable for high school students.

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