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Write the solution of the linear system corresponding to the reduced augmented matrix.

[tex]\[
\left[\begin{array}{rrr|r}
1 & 0 & 0 & -5 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0
\end{array}\right]
\][/tex]

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

A. The unique solution is [tex]\( x_1 = \boxed{-5}, x_2 = \boxed{1}, \text{ and } x_3 = \boxed{0} \)[/tex]. (Simplify your answers.)

B. The system has infinitely many solutions. The solution is [tex]\( x_1 = \boxed{ \ }, x_2 = \boxed{ \}, \text{ and } x_3 = t \)[/tex]. (Simplify your answers. Type expressions using [tex]\( t \)[/tex] as the variable.)

C. There is no solution.

Sagot :

GinnyAnsweridn
To solve the linear system corresponding to the given reduced augmented matrix:
[tex]\[ \left[\begin{array}{rrr|r} 1 & 0 & 0 & -5 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \][/tex]

we need to interpret each row of the matrix. Each row represents an equation in the system.

1. The first row is:
[tex]\[ 1x_1 + 0x_2 + 0x_3 = -5 \][/tex]
This simplifies to:
[tex]\[ x_1 = -5 \][/tex]

2. The second row is:
[tex]\[ 0x_1 + 1x_2 + 0x_3 = 1 \][/tex]
This simplifies to:
[tex]\[ x_2 = 1 \][/tex]

3. The third row is:
[tex]\[ 0x_1 + 0x_2 + 1x_3 = 0 \][/tex]
This simplifies to:
[tex]\[ x_3 = 0 \][/tex]

From these simplified equations, we see that there is a unique solution to the system:

The unique solution is:
[tex]\[ x_1 = -5, \; x_2 = 1, \; x_3 = 0 \][/tex]

So the correct choice is:

A. The unique solution is [tex]\( x_1 = -5 \)[/tex], [tex]\( x_2 = 1 \)[/tex], and [tex]\( x_3 = 0 \)[/tex].
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