https://file.aiquickdraw.com/mathbot/file/1726705344761a5rgtmeb.jpg

The system of equations provided is:

\[
\begin{aligned}
1. \ x_1 - 2x_2 + 6x_3 &= 0 \\
2. \ -4x_1 + 2x_2 - 15x_3 &= 0 \\
3. \ x_1 + 4x_2 + 8x_3 &= 0
\end{aligned}
\]

We need to determine if the system has a nontrivial solution using as few row operations as possible. This can be done using Gaussian elimination or row-reduction to find the rank of the matrix and determine if there are free variables.

Let me solve this by forming the augmented matrix and simplifying it.

\[
\begin{pmatrix}
1 & -2 & 6 & 0 \\
-4 & 2 & -15 & 0 \\
1 & 4 & 8 & 0
\end{pmatrix}
\]

I'll now proceed with row operations to put this into reduced row-echelon form.After performing Gaussian elimination, the augmented matrix becomes:

\[
\begin{pmatrix}
1 & 0 & 6.67 & 0 \\
0 & 0 & 11 & 0 \\
0 & 1 & 0.33 & 0
\end{pmatrix}
\]

This matrix shows that:

1. The third row implies \( x_2 + 0.33x_3 = 0 \), so \( x_2 = -0.33x_3 \).
2. The second row simplifies to \( 11x_3 = 0 \), implying \( x_3 = 0 \).
3. Substituting \( x_3 = 0 \) into the third row gives \( x_2 = 0 \).
4. The first row then implies \( x_1 + 6.67x_3 = 0 \), but since \( x_3 = 0 \), we get \( x_1 = 0 \).

Thus, \( x_1 = x_2 = x_3 = 0 \), indicating that the system **only has the trivial solution**.

Let me know if you would like further clarification or details on any step!

Here are 5 related questions to consider:
1. What does it mean for a system of equations to have a nontrivial solution?
2. How does Gaussian elimination help in finding solutions for linear systems?
3. What is the significance of a row of zeros in the reduced row-echelon form?
4. How can we use the rank of a matrix to determine the number of solutions to a system of equations?
5. What are free variables, and when do they occur in a linear system?

**Tip:** Always check if the system is homogeneous. Homogeneous systems (with all constant terms zero) always have at least the trivial solution.

Determine if the system has a nontrivial solution. Try to use as few row operations as possible: x₁ - 2x₂ + 6x₃ = 0, -4x₁ + 2x₂ - 15x₃ = 0, x₁ + 4x₂ + 8x₃ = 0.

Learn how to determine whether the system x₁ - 2x₂ + 6x₃ = 0, -4x₁ + 2x₂ - 15x₃ = 0, x₁ + 4x₂ + 8x₃ = 0 has a nontrivial solution by applying Gaussian elimination with minimal row operations. This solution covers key concepts such as row operations and matrix reduction.

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