$\left[ \begin{array} { r r r | r }
1 & 0 & 4 & - 2 \
0 & 1 & - 3 & 1 \
0 & 0 & 0 & 0
\end{array} \right]$
A) infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + 3 z , z = z$
B) infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + z , z = z$
C) infinitely many solutions of the form $x = - 2 - z , y = 1 + 3 z , z = z$
D) infinitely many solutions of the form $x = 1 + 3 z , y = - 2 - 4 z , z = z$

Question

$\left[ \begin{array} { r r r | r } 
1 & 0 & 4 & - 2 \
0 & 1 & - 3 & 1 \
0 & 0 & 0 & 0
\end{array} \right]$
A) infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + 3 z , z = z$
B) infinitely many solutions of the form $x = - 2 - 4 z , y = 1 + z , z = z$
C) infinitely many solutions of the form $x = - 2 - z , y = 1 + 3 z , z = z$
D) infinitely many solutions of the form $x = 1 + 3 z , y = - 2 - 4 z , z = z$

Quizplus · Accepted Answer

The given matrix is in row echelon form, representing a system of linear equations. The third row being all zeros with a zero on the right side of the equation indicates that there is not a unique solution. The system is dependent, meaning there are infinitely many solutions. The first and second rows correspond to equations $x + 4z = -2$ and $y - 3z = 1$, respectively. Solving these for $x$ and $y$ in terms of $z$ gives $x = -2 - 4z$ and $y = 1 + 3z$, with $z$ being a free variable.

[104−201−310000]\left[ \begin{array} { r r r | r } 1 & 0 & 4 & - 2 \\0 & 1 & - 3 & 1 \\0 & 0 & 0 & 0\end{array} \right]​100​010​4−30​−210​​

$\left[ \begin{array} { r r r | r } 1 & 0 & 4 & - 2 \\0 & 1 & - 3 & 1 \\0 & 0 & 0 & 0\end{array} \right]$