Find the solution of the following system using Gauss elimination. (Enter your answers as a comma-separated list.)
4x	 + 	4y	 = 	8
16x	 + 	2y	 = 	-10
(x, y) = ( 
 )
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We are given the following system of linear equations:

\[
\begin{aligned}
4x + 4y &= 8 \\
16x + 2y &= -10
\end{aligned}
\]

We will solve this system using **Gaussian elimination**. Let's go through the steps:

### Step 1: Write the augmented matrix
The system can be represented as an augmented matrix:

\[
\left[\begin{array}{cc|c}
4 & 4 & 8 \\
16 & 2 & -10
\end{array}\right]
\]

### Step 2: Row reduction
We aim to get the matrix into row echelon form.

- First, divide the first row by 4 to simplify:

\[
\left[\begin{array}{cc|c}
1 & 1 & 2 \\
16 & 2 & -10
\end{array}\right]
\]

- Next, we want to eliminate the \(x\)-term in the second row. To do this, subtract 16 times the first row from the second row:

\[
R_2 \rightarrow R_2 - 16R_1
\]

\[
\left[\begin{array}{cc|c}
1 & 1 & 2 \\
0 & -14 & -42
\end{array}\right]
\]

### Step 3: Solve for the variables
Now that we have the matrix in row echelon form, we can write the system as:

\[
\begin{aligned}
x + y &= 2 \\
-14y &= -42
\end{aligned}
\]

From the second equation, solve for \(y\):

\[
y = \frac{-42}{-14} = 3
\]

Substitute \(y = 3\) into the first equation to solve for \(x\):

\[
x + 3 = 2 \\
x = 2 - 3 = -1
\]

### Final Answer:
The solution to the system is:

\[
(x, y) = (-1, 3)
\]

Would you like more details on the Gaussian elimination process, or do you have any questions?

---

Here are 5 related questions to expand on the topic:

1. How can Gaussian elimination be extended to solve a 3x3 system of equations?
2. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
3. How would you interpret a system of equations if no solutions or infinite solutions exist?
4. How can you implement Gaussian elimination using matrices in programming languages like Python or MATLAB?
5. What are some potential pitfalls when using Gaussian elimination for larger systems?

**Tip:** Always check for pivot elements being zero before proceeding with Gaussian elimination to avoid division by zero!

Find the solution of the following system using Gauss elimination. (Enter your answers as a comma-separated list.) 4x + 4y = 8, 16x + 2y = -10 (x, y) = ( )

Learn how to solve the system of equations 4x + 4y = 8 and 16x + 2y = -10 using Gaussian elimination. This method involves row reduction and is part of linear algebra, making it suitable for Grades 9-12 students.

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