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The system of equations provided is:

\[
\begin{aligned}
1. & \quad -2x - 3y + 5z = -6 \\
2. & \quad x + 2y - 5z = 3 \\
3. & \quad -4x + y - 5z = 8 \\
\end{aligned}
\]

We will solve this system of linear equations using either substitution or elimination. Let's proceed step by step:

### Step 1: Solve the second equation for \(x\)

From equation (2):
\[
x + 2y - 5z = 3
\]
Solve for \(x\):
\[
x = 3 - 2y + 5z \quad \text{(Equation A)}
\]

### Step 2: Substitute \(x\) into the other two equations

Substitute \(x = 3 - 2y + 5z\) from Equation A into equations (1) and (3).

#### Substitution in equation (1):
\[
-2(3 - 2y + 5z) - 3y + 5z = -6
\]
Simplifying:
\[
-6 + 4y - 10z - 3y + 5z = -6
\]
\[
y - 5z = 0 \quad \text{(Equation B)}
\]

#### Substitution in equation (3):
\[
-4(3 - 2y + 5z) + y - 5z = 8
\]
Simplifying:
\[
-12 + 8y - 20z + y - 5z = 8
\]
\[
9y - 25z = 20 \quad \text{(Equation C)}
\]

### Step 3: Solve the system of two equations (B and C)

Now solve the system:
\[
\begin{aligned}
B: & \quad y - 5z = 0 \\
C: & \quad 9y - 25z = 20 \\
\end{aligned}
\]

From equation \(B\), solve for \(y\):
\[
y = 5z
\]

### Step 4: Substitute \(y = 5z\) into equation \(C\)

Substitute \(y = 5z\) into equation \(C\):
\[
9(5z) - 25z = 20
\]
Simplify:
\[
45z - 25z = 20
\]
\[
20z = 20
\]
\[
z = 1
\]

### Step 5: Solve for \(y\) and \(x\)

Now that \(z = 1\), substitute it back into the expression for \(y\):
\[
y = 5(1) = 5
\]

Finally, substitute \(y = 5\) and \(z = 1\) into Equation A to find \(x\):
\[
x = 3 - 2(5) + 5(1)
\]
\[
x = 3 - 10 + 5 = -2
\]

### Solution

The solution to the system is:
\[
x = -2, \quad y = 5, \quad z = 1
\]

Would you like further details or have any questions?

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Here are some additional related questions:

1. How can we verify the solution by substituting it back into the original equations?
2. What is the geometric interpretation of the solution of this system of equations?
3. Can this system be solved using matrix methods like Gaussian elimination?
4. How does substitution compare to elimination for solving systems of equations?
5. What happens if one of the equations is inconsistent or dependent?

**Tip:** Always verify your solutions by plugging them back into the original system to ensure they satisfy all the equations.

Solve the following system of equations for all three variables:
-2x - 3y + 5z = -6
x + 2y - 5z = 3
-4x + y - 5z = 8

Learn how to solve the system of equations -2x - 3y + 5z = -6, x + 2y - 5z = 3, and -4x + y - 5z = 8 using substitution and elimination methods. A step-by-step solution explains the process for Grades 9-12.

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