https://file.aiquickdraw.com/mathbot/file/1726965341185hum6ppv1.jpg

Let's examine the system of equations from the image, which is:

\[
\begin{aligned}
4x + y + z &= 15 \\
3x - 6y - 4z &= 22 \\
-5x + y + 2z &= -19
\end{aligned}
\]

### Step 1: Convert to Matrix Form
We can express the system as an augmented matrix:

\[
\begin{pmatrix}
4 & 1 & 1 & | & 15 \\
3 & -6 & -4 & | & 22 \\
-5 & 1 & 2 & | & -19
\end{pmatrix}
\]

### Step 2: Use Row Operations to Solve
I will now use Gaussian elimination to reduce this matrix to row-echelon form, allowing us to see if the system has a unique solution, no solution, or infinitely many solutions. I'll calculate this and provide the answer.

Let me proceed with the calculations.The system of equations has a **unique solution**, which is:

\[
x = 4, \quad y = -3, \quad z = 2
\]

Thus, the correct answer to the question is **B. Exactly 1** solution.

Would you like to see the detailed steps of the row operations used in Gaussian elimination, or any other clarification?

Here are 5 related questions to expand your understanding:
1. How does Gaussian elimination work for solving systems of equations?
2. Can systems with no solutions be identified early in Gaussian elimination?
3. What does it mean when a system has infinitely many solutions?
4. How can we use matrix inversion to solve systems of linear equations?
5. What is the difference between row-echelon form and reduced row-echelon form?

**Tip:** When solving systems of equations, always check the determinant of the coefficient matrix to verify whether a unique solution exists (non-zero determinant).

Solve the system using matrices (row operations):
4x + y + z = 15, 3x - 6y - 4z = 22, -5x + y + 2z = -19. How many solutions are there to this system?

Learn how to solve the system of linear equations 4x + y + z = 15, 3x - 6y - 4z = 22, and -5x + y + 2z = -19 using Gaussian elimination. Explore key linear algebra concepts like matrix row operations and row-echelon form to find the unique solution.

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