https://file.aiquickdraw.com/mathbot/file/1728173336931qlxwvyt3.jpg

The system of equations given in the image is:

\[
\begin{aligned}
-2x - 2y &= 16 \\
-4x - 4y &= 31
\end{aligned}
\]

To solve this system using Gaussian elimination:

### Step 1: Set up the augmented matrix
\[
\begin{pmatrix}
-2 & -2 & | & 16 \\
-4 & -4 & | & 31
\end{pmatrix}
\]

### Step 2: Perform row operations
1. The first row can be simplified by dividing by \(-2\), which gives:
   \[
   \begin{pmatrix}
   1 & 1 & | & -8 \\
   -4 & -4 & | & 31
   \end{pmatrix}
   \]

2. To eliminate the \(x\)-term from the second row, we can perform the operation:
   \[
   R_2 \rightarrow R_2 - (-4) \times R_1
   \]
   Applying this operation results in:
   \[
   \begin{pmatrix}
   1 & 1 & | & -8 \\
   0 & 0 & | & -1
   \end{pmatrix}
   \]

### Step 3: Analyze the resulting matrix
The second row is \(0 = -1\), which is a contradiction. This means that the system of equations is inconsistent and has **no solution**.

### Conclusion:
The correct answer is **No Solution**.

---

Do you want more details or have any questions? Here are five related questions to explore this topic further:

1. How does Gaussian elimination work for systems with infinitely many solutions?
2. What is the difference between consistent and inconsistent systems of equations?
3. Can a system have exactly two solutions? Why or why not?
4. How would you solve a system of three equations using Gaussian elimination?
5. What are the advantages of using Gaussian elimination over substitution?

**Tip:** Always check for consistency when using Gaussian elimination. If a row of the form \(0 = c\) appears, the system is inconsistent and has no solutions.

Use Gaussian elimination and back-substitution to solve the following system of equations. If there is a solution, write your answer in the format (x, y). -2x - 2y = 16 and -4x - 4y = 31

Learn how to solve the system of equations using Gaussian elimination and back-substitution. This problem demonstrates an inconsistent system with no solution, showcasing key algebraic techniques in linear algebra. Suitable for high school students in Grades 10-12.

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