SPLTV<div>2x+y+z=2</div><div>x-y-2z=5</div><div>x+2y+z=9</div>

To solve the system of linear equations:

\[
\begin{align*}
1. & \ \ 2x + y + z = 2 \\
2. & \ \ x - y - 2z = 5 \\
3. & \ \ x + 2y + z = 9
\end{align*}
\]

We can use methods such as substitution, elimination, or matrix-based approaches like Gaussian elimination. I'll use elimination to solve the system step by step.

### Step 1: Eliminate one variable
First, let's subtract equation 2 from equation 1 to eliminate \(y\).

\[
(2x + y + z) - (x - y - 2z) = 2 - 5
\]

Simplifying:

\[
x + 2y + 3z = -3
\]

We now have a simpler system of equations:

\[
\begin{align*}
4. & \ \ x + 2y + z = 9 \quad \text{(equation 3)} \\
5. & \ \ x + 2y + 3z = -3
\end{align*}
\]

### Step 2: Eliminate \(x\)
Next, subtract equation 4 from equation 5:

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

Simplifying:

\[
2z = -12 \quad \Rightarrow \quad z = -6
\]

### Step 3: Substitute \(z = -6\) into one of the simpler equations
Substitute \(z = -6\) into equation 4:

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

Simplifying:

\[
x + 2y - 6 = 9 \quad \Rightarrow \quad x + 2y = 15 \quad \text{(equation 6)}
\]

### Step 4: Use substitution to find \(x\) and \(y\)
Now substitute \(z = -6\) into equation 1:

\[
2x + y + (-6) = 2
\]

Simplifying:

\[
2x + y - 6 = 2 \quad \Rightarrow \quad 2x + y = 8 \quad \text{(equation 7)}
\]

We now have two equations with \(x\) and \(y\):

\[
\begin{align*}
6. & \ \ x + 2y = 15 \\
7. & \ \ 2x + y = 8
\end{align*}
\]

### Step 5: Solve for \(x\) and \(y\)
Multiply equation 6 by 2:

\[
2x + 4y = 30 \quad \text{(equation 8)}
\]

Subtract equation 7 from equation 8:

\[
(2x + 4y) - (2x + y) = 30 - 8
\]

Simplifying:

\[
3y = 22 \quad \Rightarrow \quad y = \frac{22}{3}
\]

Now substitute \(y = \frac{22}{3}\) into equation 6:

\[
x + 2 \times \frac{22}{3} = 15
\]

Simplifying:

\[
x + \frac{44}{3} = 15 \quad \Rightarrow \quad x = 15 - \frac{44}{3} = \frac{45}{3} - \frac{44}{3} = \frac{1}{3}
\]

### Final Solution:
\[
x = \frac{1}{3}, \quad y = \frac{22}{3}, \quad z = -6
\]

Would you like further explanation or a verification of the solution? Here are some related questions you might find useful:

1. How can I verify the solution using substitution?
2. What are some other methods to solve this system of equations?
3. How would the solution change if one of the constants in the system was different?
4. Can this system be solved using matrices or determinants?
5. How would I interpret this solution in a real-world context?

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

Solve the system of linear equations: 2x + y + z = 2, x - y - 2z = 5, x + 2y + z = 9

Learn how to solve the system of linear equations: 2x + y + z = 2, x - y - 2z = 5, and x + 2y + z = 9 using the elimination method. This step-by-step solution covers core algebraic concepts, including variable elimination, substitution, and consistency checks. Ideal for high school students in Grades 9-12.

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