Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{array} { r }
3 x - 2 y + 2 z - w = 2 \
4 x + y + z + 6 w = 8 \
- 3 x + 2 y - 2 z + w = 5 \
5 x + 3 z - 2 w = 1
\end{array}$$
A) $\varnothing$
B) $\left\{ \left( 2,0 , - \frac { 3 } { 37 } , \frac { 9 } { 37 } \right) \right\}$
C) $\left\{ \left( \frac { 1 } { 2 } , 0 , - \frac { 37 } { 3 } , \frac { 37 } { 9 } \right) \right\}$
D) $\left\{ \left( 1 , - \frac { 1 } { 3 } , \frac { 4 } { 9 } , 6 \right) \right\}$

Question

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{array} { r } 
3 x - 2 y + 2 z - w = 2 \
4 x + y + z + 6 w = 8 \
- 3 x + 2 y - 2 z + w = 5 \
5 x + 3 z - 2 w = 1
\end{array}$$
A) $\varnothing$
B) $\left\{ \left( 2,0 , - \frac { 3 } { 37 } , \frac { 9 } { 37 } \right) \right\}$
C) $\left\{ \left( \frac { 1 } { 2 } , 0 , - \frac { 37 } { 3 } , \frac { 37 } { 9 } \right) \right\}$
D) $\left\{ \left( 1 , - \frac { 1 } { 3 } , \frac { 4 } { 9 } , 6 \right) \right\}$

Quizplus · Accepted Answer

The system of equations is inconsistent, meaning there is no solution that satisfies all equations simultaneously. This can be determined through Gaussian elimination, which, when applied to this system, reveals a contradiction indicating that no set of values for $x$, $y$, $z$, and $w$ can satisfy all four equations at the same time.