https://file.aiquickdraw.com/mathbot/file/1726688816468axt58383.jpg

Let's analyze the system of equations and solve it step by step.

The augmented matrix \([A | b]\) is given by:

\[
\begin{pmatrix}
-1 & -2 & 2 \\
1 & 0 & 2
\end{pmatrix}
\]

We want to apply Gaussian elimination to convert this into an upper triangular matrix \(U\), and then back-substitute to find the solution \(x\).

### Step 1: Apply elimination

We can start by using row operations to create zeros below the diagonal. 

**Row 1**: \([-1 \quad -2 \quad 2]\)  
**Row 2**: \([1 \quad 0 \quad 2]\)

To eliminate the first element of Row 2, we add Row 1 to Row 2:
\[
R_2 = R_2 + R_1 = [1 + (-1) \quad 0 + (-2) \quad 2 + 2] = [0 \quad -2 \quad 4]
\]

So, the upper triangular matrix \(U\) becomes:
\[
U = \begin{pmatrix}
-1 & -2 \\
0 & -2
\end{pmatrix}
\]

The corresponding constant vector \(c\) is:
\[
c = \begin{pmatrix}
2 \\
4
\end{pmatrix}
\]

### Step 2: Back-substitution

Now we solve the system \(Ux = c\).

From the second row:
\[
-2x_2 = 4 \quad \Rightarrow \quad x_2 = \frac{4}{-2} = -2
\]

Substitute \(x_2 = -2\) into the first equation:
\[
-1x_1 - 2(-2) = 2 \quad \Rightarrow \quad -x_1 + 4 = 2
\]
\[
-x_1 = 2 - 4 = -2 \quad \Rightarrow \quad x_1 = 2
\]

Thus, the solution is:
\[
x = \begin{pmatrix}
2 \\
-2
\end{pmatrix}
\]

### Summary of results:
- \(U = \begin{pmatrix} -1 & -2 \\ 0 & -2 \end{pmatrix}\)
- \(c = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\)
- \(x = \begin{pmatrix} 2 \\ -2 \end{pmatrix}\)

Would you like more details or further clarification on any part?

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Here are 5 related questions:
1. How does Gaussian elimination work in larger matrices?
2. What is the role of back-substitution in solving linear systems?
3. Can Gaussian elimination be applied to non-square matrices?
4. How do row operations affect the determinant of a matrix?
5. What happens if a matrix becomes singular during elimination?

**Tip**: When performing Gaussian elimination, always ensure to pivot correctly if you encounter zeros in the diagonal to avoid division errors.

Apply elimination to the 2 by 3 augmented matrix [A b]. What is the triangular system Ux = c? What is the solution x?

This page walks through solving a system of linear equations represented by a 2 by 3 augmented matrix using Gaussian elimination. Learn how to convert the matrix into an upper triangular form and apply back-substitution to find the solution vector.

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