https://file.aiquickdraw.com/mathbot/file/1725452976987fbq9yl1j.jpg

The image shows a system of linear equations represented by a matrix equation:

\[
\vec{x} = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{pmatrix} x
\]

To solve this, we'll interpret the matrix and solve for the vector \(\vec{x}\).

Let's go step-by-step:

1. **Identify the matrix and vector:**

   The given matrix is:
   \[
   A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{pmatrix}
   \]
   And we want to solve for the vector \( \vec{x} \).

2. **Understand the nature of the matrix:**

   The matrix \(A\) is an upper triangular matrix, where all the elements below the main diagonal are zeros.

3. **Solve for \(\vec{x}\) using Back Substitution:**

   Let's denote the vector \( \vec{x} \) as \( \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \).

   The system of equations can be represented as:
   \[
   \begin{cases}
   x_1 + 2x_2 + 3x_3 = x_1 \\
   x_2 + 4x_3 = x_2 \\
   2x_3 = x_3
   \end{cases}
   \]

   We will solve these equations from the bottom up.

4. **Solve the third equation:**
   \[
   2x_3 = x_3
   \]

   Subtracting \(x_3\) from both sides:
   \[
   x_3 = 0
   \]

5. **Solve the second equation:**
   \[
   x_2 + 4x_3 = x_2
   \]

   Since \(x_3 = 0\), the equation simplifies to:
   \[
   x_2 = x_2
   \]

   This means \(x_2\) can be any real number. Let's denote it as a free parameter \( t \).

6. **Solve the first equation:**
   \[
   x_1 + 2x_2 + 3x_3 = x_1
   \]

   Substituting \(x_3 = 0\) and simplifying:
   \[
   x_1 + 2x_2 = x_1
   \]

   Subtracting \(x_1\) from both sides:
   \[
   2x_2 = 0
   \]

   Since \(x_2 = t\), we get \(2t = 0\), so \(t = 0\).

7. **Conclusion:**

   The solution to the system is:

   \[
   \vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
   \]

This means the only solution for this system is the **zero vector**.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What is an upper triangular matrix, and what are its properties?
2. How does back substitution work in solving linear systems?
3. What is the determinant of an upper triangular matrix?
4. What does it mean if a system of linear equations has only the trivial solution?
5. How can matrix row operations help in solving a linear system?

**Tip:** When dealing with matrices, recognizing their structure (like upper triangular, diagonal, etc.) can simplify solving them greatly.

Solve the following system of linear equations represented by the matrix equation.

Explore the step-by-step process of solving a system of linear equations represented by an upper triangular matrix. This solution covers linear algebra concepts including matrix multiplication, back substitution, and the properties of upper triangular matrices. Suitable for undergraduate students.

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