Write augmented the matrix corresponding to the following system of linear equations.
negative 3 x 1
plus
6 x 2
plus
7 x 3
equals
4
negative 9 x 1
plus
5 x 2
minus
x 3
equals
7
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Part 1
What is the corresponding augmented​ matrix? Choose the correct choice below and fill in the answer boxes to complete your answer.
A.
The augmented matrix is left bracket Start 2 By 4 Table 1st Row 1st Column nothing 2nd Column nothing 3rd Column nothing 4st Column nothing 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing 4st Column nothing EndTable right bracket
  
  
  
  
  
  
  
  
 .
B.
The augmented matrix is left bracket Start 3 By 3 Table 1st Row 1st Column nothing 2nd Column nothing 3rd Column nothing 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing 3rd Row 1st Column nothing 2nd Column nothing 3rd Column nothing EndTable right bracket
  
  
  
  
  
  
  
  
  
 .

Let's write the augmented matrix for the given system of linear equations.

The system of equations is:

\[
\begin{aligned}
-3x_1 + 6x_2 + 7x_3 &= 4 \\
-9x_1 + 5x_2 - x_3 &= 7
\end{aligned}
\]

To create the augmented matrix, we include the coefficients of the variables and the constants on the right-hand side:

\[
\begin{bmatrix}
-3 & 6 & 7 & | & 4 \\
-9 & 5 & -1 & | & 7
\end{bmatrix}
\]

In a more compact form without the vertical bar, the augmented matrix is:

\[
\begin{bmatrix}
-3 & 6 & 7 & 4 \\
-9 & 5 & -1 & 7
\end{bmatrix}
\]

So, the correct answer choice corresponds to:

**Choice A:**

The augmented matrix is:

\[
\left[\begin{array}{cccc}
-3 & 6 & 7 & 4 \\
-9 & 5 & -1 & 7 \\
\end{array}\right]
\]

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Would you like more details or have any questions?

Here are five related questions:
1. How do you solve a system of linear equations using the augmented matrix?
2. What does it mean if the augmented matrix has a row of zeros?
3. How do you find the inverse of a matrix?
4. What is the role of row operations in solving linear systems?
5. How can you determine if a system of equations has no solution using an augmented matrix?

**Tip:** When working with augmented matrices, always align the matrix entries with their corresponding variables to avoid errors in the setup.

Learn how to construct the augmented matrix for a system of linear equations. This example covers two equations involving variables x₁, x₂, and x₃. Suitable for high school students studying linear algebra concepts.

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