The matrix is the reduced echelon matrix for a system with variables x 1 , x 2 , x 3 , and x 4 . Find the solution set of the system. $$\left[ \begin{array} { r r r r | r }
1 & 0 & 0 & 10 & - 8 \
0 & 1 & 0 & 0 & 5 \
0 & 0 & 1 & 6 & 3 \
0 & 0 & 0 & 0 & 0
\end{array} \right]$$
A) ( $-$ 8 $-$ 10 k , 5, 3 $-$ 6 k , k ), where k is any real number
B) ( $-$ 8 $-$ 10 k , 5, 3 + 6 k , k ), where k is any real number
C) ( $-$ 8 + 10 k , 5, 3 $-$ 6 k , k ), where k is any real number
D) ( $-$ 8 + 10 k , 5, 3 + 6 k , k ), where k is any real number
E) inconsistent

Question

The matrix is the reduced echelon matrix for a system with variables x 1 , x 2 , x 3 , and x 4 . Find the solution set of the system. $$\left[ \begin{array} { r r r r | r } 
1 & 0 & 0 & 10 & - 8 \
0 & 1 & 0 & 0 & 5 \
0 & 0 & 1 & 6 & 3 \
0 & 0 & 0 & 0 & 0
\end{array} \right]$$
A) ( $-$ 8 $-$ 10 k , 5, 3 $-$ 6 k , k ), where k is any real number
B) ( $-$ 8 $-$ 10 k , 5, 3 + 6 k , k ), where k is any real number
C) ( $-$ 8 + 10 k , 5, 3 $-$ 6 k , k ), where k is any real number
D) ( $-$ 8 + 10 k , 5, 3 + 6 k , k ), where k is any real number
E) inconsistent

Quizplus · Accepted Answer

The given matrix corresponds to the system of equations: $x_1 + 10x_4 = -8$, $x_2 = 5$, $x_3 + 6x_4 = 3$, with $x_4$ being a free variable (let $x_4 = k$). Solving for $x_1$ and $x_3$ in terms of $k$ gives $x_1 = -8 - 10k$ and $x_3 = 3 - 6k$, hence the solution is $(-8 - 10k, 5, 3 - 6k, k)$.

The Matrix Is the Reduced Echelon Matrix for a System [10010−8010050016300000]\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 10 & - 8 \\0 & 1 & 0 & 0 & 5 \\0 & 0 & 1 & 6 & 3 \\0 & 0 & 0 & 0 & 0\end{array} \right]​1000​0100​0010​10060​−8530​​

The Matrix Is the Reduced Echelon Matrix for a System $\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 10 & - 8 \\0 & 1 & 0 & 0 & 5 \\0 & 0 & 1 & 6 & 3 \\0 & 0 & 0 & 0 & 0\end{array} \right]$