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To solve part (a) of the system of equations, let's rewrite the equations for clarity:

1. \( x + y + z = 2 \)
2. \( 4x + 2y + z = 1 \)
3. \( 9x + 3y + z = 8 \)

### Step 1: Eliminate \( z \) from two pairs of equations.
We subtract the first equation from the second and third equations:

From equation (2) - equation (1):
\[
(4x + 2y + z) - (x + y + z) = 1 - 2
\]
\[
3x + y = -1 \tag{4}
\]

From equation (3) - equation (1):
\[
(9x + 3y + z) - (x + y + z) = 8 - 2
\]
\[
8x + 2y = 6 \tag{5}
\]

### Step 2: Solve the system of two equations in two variables (\(x\) and \(y\)).
Equations (4) and (5) are:
1. \( 3x + y = -1 \)
2. \( 8x + 2y = 6 \)

Multiply equation (4) by 2 to align coefficients of \(y\):
\[
2(3x + y) = 2(-1)
\]
\[
6x + 2y = -2 \tag{6}
\]

Subtract equation (6) from equation (5):
\[
(8x + 2y) - (6x + 2y) = 6 - (-2)
\]
\[
2x = 8
\]
\[
x = 4
\]

Substitute \(x = 4\) into equation (4):
\[
3(4) + y = -1
\]
\[
12 + y = -1
\]
\[
y = -13
\]

### Step 3: Solve for \(z\).
Substitute \(x = 4\) and \(y = -13\) into the first equation:
\[
x + y + z = 2
\]
\[
4 - 13 + z = 2
\]
\[
-9 + z = 2
\]
\[
z = 11
\]

### Final Solution:
\[
x = 4, \, y = -13, \, z = 11
\]

Would you like a detailed explanation of any step or help with part (b)? 

### Related Questions:
1. How would this approach change for systems with more variables?
2. What are alternative methods for solving systems of equations (e.g., matrix methods)?
3. How can graphing help visualize solutions to such systems?
4. What is the significance of consistency in a system of equations?
5. How does elimination compare to substitution in terms of solving efficiency?

### Tip:
Always verify solutions by substituting back into the original equations to ensure accuracy!

Solve the following system of equations: (a) x + y + z = 2, 4x + 2y + z = 1, 9x + 3y + z = 8.

Learn how to solve a system of three linear equations using the elimination method. The equations are x + y + z = 2, 4x + 2y + z = 1, and 9x + 3y + z = 8. This solution explains key algebraic concepts and techniques, suitable for high school students.

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