Solve using augmented matrix methods. Use a graphing calculator to perform the row operations.
0.4x 1plus
3.97x 2equals
6
2.5x 1plus
5.62x 2equals
8
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The unique solution is x Subscript 1equals
  
enter your response here and x Subscript 2equals
  
enter your response here. 
​(Round to four decimal places as​ needed.)
B.
The system has infinitely many solutions. The solution is x 1equals
  
enter your response here and x 2equalst.
​(Simplify your answer. Type an expression using t as the​ variable.)
C.
There is no solution.

Let's solve the system of equations using augmented matrix methods.

The system is:

\[
0.4x_1 + 3.97x_2 = 6
\]
\[
2.5x_1 + 5.62x_2 = 8
\]

### Step 1: Set up the augmented matrix
The augmented matrix representing this system is:

\[
\begin{pmatrix}
0.4 & 3.97 & | & 6 \\
2.5 & 5.62 & | & 8
\end{pmatrix}
\]

### Step 2: Perform row operations
We'll use Gaussian elimination to solve this. The goal is to get the matrix into row echelon form and solve for \(x_1\) and \(x_2\).

#### Step 2.1: Eliminate the first term of the second row.
To eliminate the \(2.5\) in the second row, we need to create a leading 1 in the first row first. We divide the first row by 0.4:

\[
\begin{pmatrix}
1 & 9.925 & | & 15 \\
2.5 & 5.62 & | & 8
\end{pmatrix}
\]

Now, subtract \(2.5\) times the first row from the second row to eliminate the leading coefficient in the second row:

\[
R_2 = R_2 - 2.5 \cdot R_1
\]

\[
\begin{pmatrix}
1 & 9.925 & | & 15 \\
0 & -18.6875 & | & -29.5
\end{pmatrix}
\]

#### Step 2.2: Solve for \(x_2\)
Now that the second row only has one variable, we can solve for \(x_2\). Divide the second row by \(-18.6875\):

\[
x_2 = \frac{-29.5}{-18.6875} \approx 1.5787
\]

#### Step 2.3: Back-substitute to find \(x_1\)
Now substitute \(x_2 = 1.5787\) into the first equation to solve for \(x_1\):

\[
x_1 + 9.925 \times 1.5787 = 15
\]

\[
x_1 + 15.656 = 15
\]

\[
x_1 = 15 - 15.656 \approx -0.656
\]

### Final Answer:
The unique solution to the system is:
\[
x_1 \approx -0.656 \quad \text{and} \quad x_2 \approx 1.5787
\]

This corresponds to option **A**.

---

Do you want more details or have any questions about this?  
Here are some related questions to expand on this:

1. How can we verify the solution by substituting \(x_1\) and \(x_2\) back into the original system?
2. What are the properties of a system of equations that lead to a unique solution?
3. How does row echelon form help in solving systems of equations?
4. Can a system have no solutions or infinite solutions, and how would the augmented matrix reflect that?
5. What are some common errors to avoid when performing row operations?

**Tip**: Always check your final solution by substituting the values back into the original equations to ensure accuracy.

Solve using augmented matrix methods. Use a graphing calculator to perform the row operations.
0.4x1 + 3.97x2 = 6
2.5x1 + 5.62x2 = 8
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A. The unique solution is x1 = (enter your response here) and x2 = (enter your response here). ​(Round to four decimal places as​ needed.)
B. The system has infinitely many solutions. The solution is x1 = (enter your response here) and x2 = t. ​(Simplify your answer. Type an expression using t as the​ variable.)
C. There is no solution.

Learn how to solve a system of linear equations using augmented matrix methods and Gaussian elimination. This problem demonstrates key linear algebra concepts and techniques like row operations and back substitution. Suitable for advanced high school and college-level students.

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