Let's examine the augmented matrix you provided:
[tex]\[
\left[\begin{array}{rrr|r}
1 & -\frac{1}{3} & -\frac{1}{3} & \frac{13}{3} \\
0 & 1 & -\frac{5}{2} & \frac{7}{2} \\
0 & 0 & 1 & -3
\end{array}\right]
\][/tex]
We aim to interpret this in context. This augmented matrix represents the system of equations:
1. \( x_1 - \frac{1}{3}x_2 - \frac{1}{3}x_3 = \frac{13}{3} \)
2. \( x_4 - 2.5x_5 = 3.5 \)
3. \( x_6 = -3 \)
We can rewrite this as:
1. \( x_1 - \frac{1}{3}x_2 - \frac{1}{3}x_3 = \frac{13}{3} \)
2. \( x_4 - 2.5x_5 = 3.5 \)
3. \( x_6 = -3 \)
The matrix is in echelon form (upper triangular form), which is very useful for finding solutions to linear systems.
Let's interpret the entries in the matrix step-by-step:
### Row 1:
The first row is:
[tex]\[
[1, -0.3333333333333333, -0.3333333333333333, 4.333333333333333]
\][/tex]
This corresponds to the equation:
[tex]\[
x_1 - \frac{1}{3}x_2 - \frac{1}{3}x_3 = \frac{13}{3}
\][/tex]
Or, if we convert the fractions to decimals:
[tex]\[
x_1 - 0.3333333333333333x_2 - 0.3333333333333333x_3 = 4.333333333333333
\][/tex]
### Row 2:
The second row is:
[tex]\[
[0, 1, -2.5, 3.5]
\][/tex]
This corresponds to the equation:
[tex]\[
x_4 - 2.5x_5 = 3.5
\][/tex]
### Row 3:
The third row is:
[tex]\[
[0, 0, 1, -3]
\][/tex]
This corresponds to the equation:
[tex]\[
x_6 = -3
\][/tex]
### Recap:
So, the system of equations derived from the provided augmented matrix is:
1. \( x_1 - 0.3333x_2 - 0.3333x_3 = 4.3333 \)
2. \( x_4 - 2.5x_5 = 3.5 \)
3. \( x_6 = -3 \)
This is a detailed breakdown of how we interpret the given augmented matrix into corresponding system of linear equations. Each row of the augmented matrix gives us one of these equations, which collectively represent the solution space for a set of variables.
