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To solve the given system of linear equations using either Gaussian elimination or Gauss-Jordan elimination, let's write down the equations in matrix form and perform the elimination steps.

The system of equations is:

\[
\begin{align*}
4x + 12y - 7z - 20w &= 24, \\
3x + 9y - 5z - 28w &= 30.
\end{align*}
\]

We'll convert these equations into an augmented matrix and proceed with Gaussian elimination.

The augmented matrix of the system is:

\[
\begin{bmatrix}
4 & 12 & -7 & -20 & | & 24 \\
3 & 9 & -5 & -28 & | & 30
\end{bmatrix}
\]

Let's perform row operations to reduce this matrix to row-echelon form and then solve the system.The row reduction resulted in invalid values (NaN and infinity), which indicates that the system of equations may be inconsistent or has some dependency issues, leading to no clear unique solution. Let's analyze this more carefully:

Upon reaching a point where rows contain invalid entries (such as zeros in positions that should be pivots or conflicting results), it suggests that:

1. **No Solution**: The equations are contradictory, meaning the system has no common intersection point.
2. **Infinite Solutions**: The rows might be dependent but produce undefined or conflicting information.

In this case, the computation hints at a contradictory system, suggesting **no solution**. Let me know if you want more clarification or additional questions. 

### Relative Questions:
1. What is Gaussian elimination and how does it work?
2. How can we detect an inconsistent system using row reduction?
3. What are the differences between Gaussian elimination and Gauss-Jordan elimination?
4. Can we solve inconsistent systems using other methods?
5. How can we express solutions in terms of parameters for consistent but underdetermined systems?

**Tip:** Always check for row dependencies and contradictions when reducing matrices to avoid misinterpretations.

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x, y, z, and w in terms of the parameters t and s.) 4x + 12y − 7z − 20w = 24; 3x + 9y − 5z − 28w = 30.

Learn how to solve a system of linear equations using Gaussian elimination and Gauss-Jordan elimination methods. This problem involves row reduction techniques, consistency checks, and determining the solution set. Suitable for high school students in Grades 10-12.

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