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Let's solve the system of equations by the method of elimination (or reduction) to find the values of \( x \), \( y \), and \( z \).

Given system:

1. \( 3x - 6z = 18 \)
2. \( x - 3y - 3z = 29 \)
3. \( x + y - 3z = 1 \)
4. \( 4x + y + z = -1 \)

### Step 1: Simplify Equations if Possible
From equation (1):
\[
3x - 6z = 18
\]
Divide by 3:
\[
x - 2z = 6 \quad \text{(Equation A)}
\]

### Step 2: Use Equation A to Substitute and Eliminate Variables
Using \( x = 6 + 2z \) from Equation A, substitute \( x \) in the remaining equations (2), (3), and (4) to eliminate \( x \).

#### Substitute into Equation (2):
\[
(6 + 2z) - 3y - 3z = 29
\]
Simplify:
\[
6 - 3y - z = 29
\]
\[
-3y - z = 23 \quad \text{(Equation B)}
\]

#### Substitute into Equation (3):
\[
(6 + 2z) + y - 3z = 1
\]
Simplify:
\[
6 + y - z = 1
\]
\[
y - z = -5 \quad \text{(Equation C)}
\]

#### Substitute into Equation (4):
\[
4(6 + 2z) + y + z = -1
\]
\[
24 + 8z + y + z = -1
\]
Combine terms:
\[
y + 9z = -25 \quad \text{(Equation D)}
\]

### Step 3: Solve Equations B, C, and D
Now we have three equations in terms of \( y \) and \( z \):

1. \( -3y - z = 23 \) (Equation B)
2. \( y - z = -5 \) (Equation C)
3. \( y + 9z = -25 \) (Equation D)

#### Solve Equation B and C
From Equation C:
\[
y = -5 + z
\]
Substitute \( y = -5 + z \) into Equation B:
\[
-3(-5 + z) - z = 23
\]
\[
15 - 3z - z = 23
\]
\[
15 - 4z = 23
\]
\[
-4z = 8
\]
\[
z = -2
\]

#### Substitute \( z = -2 \) into Equation C to Find \( y \)
\[
y = -5 + (-2)
\]
\[
y = -7
\]

#### Substitute \( y = -7 \) and \( z = -2 \) into Equation A to Find \( x \)
\[
x - 2(-2) = 6
\]
\[
x + 4 = 6
\]
\[
x = 2
\]

### Solution
The solution to the system is:
\[
x = 2, \quad y = -7, \quad z = -2
\]

Would you like further details or have any questions?

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### Related Questions
1. What is the elimination (reduction) method in solving systems of equations?
2. How do substitution and elimination methods compare in solving systems?
3. Can all systems of equations be solved by elimination? Why or why not?
4. What does it mean if a system of equations has no solution?
5. How would the solution change if one of the constants in the system was altered?

### Tip
Always check your solution by substituting back into the original equations to ensure accuracy.

Solve the following system by the method of reduction:

3x - 6z = 18
x - 3y - 3z = 29
x + y - 3z = 1
4x + y + z = -1

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

Explore the process of solving a system of equations with three variables using the elimination method. This problem covers algebraic concepts, including substitution and elimination techniques, suitable for high school students in Grades 9-12.

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