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. $$\left[ \begin{array} { l l l l | r }
1 & 0 & 0 & 0 & - 3 \
0 & 1 & 0 & 0 & - 3 \
0 & 0 & 1 & 0 & 15 \
0 & 0 & 0 & 1 & - 6
\end{array} \right]$$

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. $$\left[ \begin{array} { l l l l | r } 
1 & 0 & 0 & 0 & - 3 \
0 & 1 & 0 & 0 & - 3 \
0 & 0 & 1 & 0 & 15 \
0 & 0 & 0 & 1 & - 6
\end{array} \right]$$

Quizplus · Accepted Answer

The reduced echelon form indicates that the system is already in row echelon form, meaning that each row has a leading coefficient (first non-zero entry) that is to the right of the leading coefficient of the previous row. Moreover, each pivot column (column with the leading coefficient) has zeros everywhere else. Thus, we can read off the system of equations directly from the reduced echelon form:

\begin{alignat*}{6}
x_1\qquad&\qquad && && &&= -3 \
&& x_2 \qquad & \qquad && &&= -3 \
&& && x_3 \qquad &&= 15 \
&& && && x_4 &= -6
\end{alignat*}

We see that this system has a unique solution: $(x_1, x_2, x_3, x_4) = (-3, -3, 15, -6)$. Therefore, the correct answer is choice $\boxed{	extbf{(A)}}$.

The Matrix Is the Reduced Echelon Matrix for a System [1000−30100−30010150001−6]\left[ \begin{array} { l l l l | r } 1 & 0 & 0 & 0 & - 3 \\0 & 1 & 0 & 0 & - 3 \\0 & 0 & 1 & 0 & 15 \\0 & 0 & 0 & 1 & - 6\end{array} \right]​1000​0100​0010​0001​−3−315−6​​

The Matrix Is the Reduced Echelon Matrix for a System $\left[ \begin{array} { l l l l | r } 1 & 0 & 0 & 0 & - 3 \\0 & 1 & 0 & 0 & - 3 \\0 & 0 & 1 & 0 & 15 \\0 & 0 & 0 & 1 & - 6\end{array} \right]$