The augmented matrix is given for a system of equations. If the system is consistent, find the general solution.
Otherwise state that there is no solution.
-$$\left[ \begin{array} { r r r r }
1 & 0 & 6 & 3 \
0 & 1 & - 2 & - 3 \
0 & 0 & 0 & 0
\end{array} \right]$$
A) $x _ { 1 } = 3 - 6 x _ { 3 }$
$x _ { 2 } = - 3 + 2 x _ { 3 }$
$x _ { 3 } = 0$

B) $x _ { 1 } = 3 - 6 x _ { 3 }$
$x _ { 2 }$ is free
$x _ { 3 } = \frac { 3 } { 2 } + \frac { 1 } { 2 } x _ { 2 }$

C) $x _ { 1 } = 3 - 6 x _ { 3 }$
$x _ { 2 } = - 3 + 2 x _ { 3 }$
$\mathrm { x } _ { 3 }$ is free

D) No solution

Question

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution.
Otherwise state that there is no solution.
-$$\left[ \begin{array} { r r r r } 
1 & 0 & 6 & 3 \
0 & 1 & - 2 & - 3 \
0 & 0 & 0 & 0
\end{array} \right]$$
A) $x _ { 1 } = 3 - 6 x _ { 3 }$
$x _ { 2 } = - 3 + 2 x _ { 3 }$
$x _ { 3 } = 0$

B) $x _ { 1 } = 3 - 6 x _ { 3 }$
$x _ { 2 }$ is free
$x _ { 3 } = \frac { 3 } { 2 } + \frac { 1 } { 2 } x _ { 2 }$

C) $x _ { 1 } = 3 - 6 x _ { 3 }$
$x _ { 2 } = - 3 + 2 x _ { 3 }$
$\mathrm { x } _ { 3 }$ is free

D) No solution

Quizplus · Accepted Answer

The given augmented matrix corresponds to a system of equations that is consistent because there is no row of the form [0 0 0 | b] with $b \neq 0$. The system has a free variable ($x_3$) since there are more variables (3) than non-zero rows (2) in the reduced row-echelon form. The solutions for $x_1$ and $x_2$ are expressed in terms of $x_3$, making $x_3$ the free variable.

The Augmented Matrix Is Given for a System of Equations [106301−2−30000]\left[ \begin{array} { r r r r } 1 & 0 & 6 & 3 \\0 & 1 & - 2 & - 3 \\0 & 0 & 0 & 0\end{array} \right]​100​010​6−20​3−30​​

The Augmented Matrix Is Given for a System of Equations $\left[ \begin{array} { r r r r } 1 & 0 & 6 & 3 \\0 & 1 & - 2 & - 3 \\0 & 0 & 0 & 0\end{array} \right]$