solve using gaussian elimination method: a+2b+c=3,2a+2b-2c=-4,-a+b+3c=6

To solve the system of linear equations using the Gaussian elimination method, we will proceed with the following steps:

Given the system of equations:
\[
\begin{aligned}
1. & \quad a + 2b + c = 3 \\
2. & \quad 2a + 2b - 2c = -4 \\
3. & \quad -a + b + 3c = 6
\end{aligned}
\]

### Step 1: Write the system as an augmented matrix.
\[
\begin{pmatrix}
1 & 2 & 1 & | & 3 \\
2 & 2 & -2 & | & -4 \\
-1 & 1 & 3 & | & 6
\end{pmatrix}
\]

### Step 2: Perform row operations to obtain a row echelon form.

1. **R2 → R2 - 2R1** (Eliminate the first element of the second row)
\[
\begin{pmatrix}
1 & 2 & 1 & | & 3 \\
0 & -2 & -4 & | & -10 \\
-1 & 1 & 3 & | & 6
\end{pmatrix}
\]

2. **R3 → R3 + R1** (Eliminate the first element of the third row)
\[
\begin{pmatrix}
1 & 2 & 1 & | & 3 \\
0 & -2 & -4 & | & -10 \\
0 & 3 & 4 & | & 9
\end{pmatrix}
\]

3. **R3 → R3 + \frac{3}{2}R2** (Eliminate the second element of the third row)
\[
\begin{pmatrix}
1 & 2 & 1 & | & 3 \\
0 & -2 & -4 & | & -10 \\
0 & 0 & -2 & | & -6
\end{pmatrix}
\]

### Step 3: Back-substitution to find the solutions.

1. From the third row: \(-2c = -6 \Rightarrow c = 3\).
2. From the second row: \(-2b - 4(3) = -10 \Rightarrow -2b - 12 = -10 \Rightarrow -2b = 2 \Rightarrow b = -1\).
3. From the first row: \(a + 2(-1) + 3 = 3 \Rightarrow a - 2 + 3 = 3 \Rightarrow a + 1 = 3 \Rightarrow a = 2\).

### Solution:
\[
\boxed{a = 2, \, b = -1, \, c = 3}
\]

Would you like a more detailed explanation or have any questions?

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Here are 8 follow-up questions:
1. How does Gaussian elimination differ from the Gauss-Jordan method?
2. What is the significance of the augmented matrix in solving systems of equations?
3. How would the solution change if the second equation was altered slightly?
4. Can you solve a system of linear equations using matrix inversion?
5. What does it mean if the augmented matrix has a row of all zeros?
6. How do you identify if a system of equations has no solution or infinitely many solutions using Gaussian elimination?
7. What other methods can be used to solve systems of linear equations?
8. How would you solve a system of equations with more variables than equations?

**Tip:** Always double-check your row operations during Gaussian elimination to avoid calculation errors.

Learn how to solve a system of linear equations using the Gaussian elimination method with a step-by-step explanation. This problem covers key concepts in linear algebra, including augmented matrix representation and row operations. Suitable for advanced high school students.

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