https://file.aiquickdraw.com/mathbot/file/1726538279886cuy1w1mp.jpg

The system of equations in your image is:

\[
\begin{aligned}
-2x - y + 2z &= -5 \\
-3x + y + 10z &= -7 \\
-6x + 3y + 8z &= 1
\end{aligned}
\]

Let's solve this system step-by-step using either substitution or elimination. I will proceed with the elimination method.

### Step 1: Eliminate \( y \)
We will first eliminate \( y \) from two pairs of equations.

**Equation 1:** \( -2x - y + 2z = -5 \)

**Equation 2:** \( -3x + y + 10z = -7 \)

**Equation 3:** \( -6x + 3y + 8z = 1 \)

We add **Equation 1** and **Equation 2** to eliminate \( y \):

\[
(-2x - y + 2z) + (-3x + y + 10z) = -5 + (-7)
\]
\[
-5x + 12z = -12
\]
\[
\text{Simplifying: } -5x + 12z = -12
\]
Divide by -1:
\[
5x - 12z = 12 \quad \text{(Equation 4)}
\]

Next, we eliminate \( y \) between **Equation 2** and **Equation 3**. Multiply **Equation 2** by 3 to align the coefficients of \( y \):

\[
3(-3x + y + 10z) = 3(-7)
\]
\[
-9x + 3y + 30z = -21 \quad \text{(Equation 5)}
\]

Now subtract **Equation 3** from **Equation 5** to eliminate \( y \):

\[
(-9x + 3y + 30z) - (-6x + 3y + 8z) = -21 - 1
\]
\[
-3x + 22z = -22
\]
\[
\text{Simplifying: } -3x + 22z = -22 \quad \text{(Equation 6)}
\]

### Step 2: Solve the system of two equations with \( x \) and \( z \)

We now have two equations:

1. \( 5x - 12z = 12 \)  (Equation 4)
2. \( -3x + 22z = -22 \)  (Equation 6)

Solve this system using elimination or substitution.

Multiply **Equation 1** by 3 and **Equation 2** by 5 to align the \( x \) terms:

\[
3(5x - 12z) = 3(12) \quad \Rightarrow \quad 15x - 36z = 36
\]
\[
5(-3x + 22z) = 5(-22) \quad \Rightarrow \quad -15x + 110z = -110
\]

Now add these two equations:

\[
(15x - 36z) + (-15x + 110z) = 36 + (-110)
\]
\[
74z = -74
\]
\[
z = -1
\]

### Step 3: Solve for \( x \)

Substitute \( z = -1 \) into **Equation 4**:

\[
5x - 12(-1) = 12
\]
\[
5x + 12 = 12
\]
\[
5x = 0
\]
\[
x = 0
\]

### Step 4: Solve for \( y \)

Substitute \( x = 0 \) and \( z = -1 \) into any of the original equations. Using **Equation 1**:

\[
-2(0) - y + 2(-1) = -5
\]
\[
0 - y - 2 = -5
\]
\[
-y = -3
\]
\[
y = 3
\]

### Final Answer:
\[
x = 0, \quad y = 3, \quad z = -1
\]

Would you like a more detailed explanation of any step? Here are 5 related questions you can explore:

1. What is the elimination method and how does it work for solving systems of equations?
2. How can you solve a system of three variables using substitution?
3. What are some other methods to solve systems of linear equations?
4. How does Gaussian elimination differ from the method used here?
5. How can systems of linear equations be solved using matrices?

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

Solve the following system of equations for all three variables: -2x - y + 2z = -5, -3x + y + 10z = -7, -6x + 3y + 8z = 1

Learn how to solve a complex system of three variables using linear equations. This guide offers a comprehensive walkthrough using elimination and substitution methods, ideal for high school or college students.

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