Write the system of linear equations that is represented by the following augmented matrix. Assume that the variables are x 1​, x 2​, ....
left bracket Start 3 By 4 Table 1st Row 1st Column 2 2nd Column negative 3 3rd Column 0 4st Column 7 2nd Row 1st Column 0 2nd Column 7 3rd Column negative 4 4st Column 1 3rd Row 1st Column negative 6 2nd Column 0 3rd Column 3 4st Column 5 EndTable right bracket
 
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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.)

The augmented matrix is:

\[
\begin{pmatrix}
2 & -3 & 0 & 7 \\
0 & 7 & -4 & 1 \\
-6 & 0 & 3 & 5
\end{pmatrix}
\]

This matrix represents a system of linear equations, where each row corresponds to one equation, and the last column represents the constants on the right-hand side of the equation.

### To write the equation corresponding to the first row:
- The first row is \( (2, -3, 0, 7) \), which represents:
  \[
  2x_1 - 3x_2 + 0x_3 = 7
  \]
- Simplifying the equation:
  \[
  2x_1 - 3x_2 = 7
  \]

Thus, the equation corresponding to the first row is:
\[
2x_1 - 3x_2 = 7
\]

Would you like me to continue with the other rows, or explain any steps in more detail?

### 5 Related Questions:
1. What is the equation for the second row of this matrix?
2. How can this system of equations be solved using substitution or elimination?
3. What is the meaning of the augmented matrix in terms of linear systems?
4. Can you represent the system as a matrix equation \( A\mathbf{x} = \mathbf{b} \)?
5. How do you interpret solutions when a system has no solution, one solution, or infinitely many solutions?

### Tip:
When converting an augmented matrix to equations, always remember that the columns correspond to the variables, and the last column represents the constants on the right-hand side.

Write the system of linear equations that is represented by the following augmented matrix. Assume that the variables are x₁, x₂, ....
[ [2, -3, 0, 7], [0, 7, -4, 1], [-6, 0, 3, 5] ]
What equation corresponds to the first row of the matrix?

Learn how to write the system of linear equations represented by an augmented matrix. This problem involves understanding the connection between matrix rows and linear equations, specifically for variables x₁, x₂, and x₃. Suitable for students in Grades 9-12.

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