Write the system of linear equations that is represented by the following augmented matrix. Assume that the variables are 

x 1x1​, 

x 2x2​, 

....

left bracket Start 3 By 4 Table 1st Row 1st Column 3 2nd Column negative 7 3rd Column 0 4st Column 2 2nd Row 1st Column 8 2nd Column 6 3rd Column negative 2 4st Column 0 3rd Row 1st Column negative 7 2nd Column 0 3rd Column 4 4st Column 3 EndTable right bracket

3

−7

0

2

8

6

−2

0

−7

0

4

3

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Part 1

What equation corresponds to the first row of the​ matrix?   

enter your response here

​(Type your answer in standard​ form.)

To write the system of linear equations corresponding to the given augmented matrix, we need to interpret each row of the matrix as an equation.

The augmented matrix is:

\[
\left[\begin{array}{ccc|c}
3 & -7 & 0 & 2 \\
8 & 6 & -2 & 0 \\
-7 & 0 & 4 & 3 \\
\end{array}\right]
\]

Assuming the variables are \( x_1, x_2, \) and \( x_3 \), we can write the system of equations as follows:

### First Row:
\[
3x_1 - 7x_2 + 0x_3 = 2
\]

Simplifying the equation (since \( 0x_3 \) cancels out):

\[
3x_1 - 7x_2 = 2
\]

### Second Row:
\[
8x_1 + 6x_2 - 2x_3 = 0
\]

### Third Row:
\[
-7x_1 + 0x_2 + 4x_3 = 3
\]

Simplifying the equation (since \( 0x_2 \) cancels out):

\[
-7x_1 + 4x_3 = 3
\]

So, the system of linear equations corresponding to the matrix is:

\[
\begin{aligned}
1.\ &3x_1 - 7x_2 = 2 \\
2.\ &8x_1 + 6x_2 - 2x_3 = 0 \\
3.\ &-7x_1 + 4x_3 = 3
\end{aligned}
\]

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Would you like more details or have any questions?

Here are five related questions:
1. How do you solve this system of equations using substitution or elimination?
2. What is the significance of each row in the augmented matrix?
3. How would the system change if the constant terms were different?
4. What happens if the coefficient of a variable is zero in an equation?
5. How can you determine if this system of equations has a unique solution?

**Tip:** When converting an augmented matrix to a system of equations, always match the coefficients to the correct variable and remember to simplify where possible.

Learn how to write a system of linear equations from an augmented matrix with a detailed step-by-step solution. This problem covers key concepts in algebra, including systems of linear equations and matrix operations. Suitable for high school students.

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