Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
-$$\begin{array} { r }
x + y + z + w = 8 \
3 x + 2 y + z + 4 w = 21 \
4 x + 4 y + 5 z + 8 w = 30 \
2 x + 3 y + 6 z + 9 w = 15
\end{array}$$
A) $\{ ( - 6 w + 3,9 w + 7 , - 4 w - 2 , w ) \}$
B) $\{ ( 5 w + 11 , - 3 w - 7 , - 3 w + 4 , w ) \}$
C) $\{ ( - 3,16 , - 6,1 ) \}$
D) $\varnothing$

Question

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
-$$\begin{array} { r } 
x + y + z + w = 8 \
3 x + 2 y + z + 4 w = 21 \
4 x + 4 y + 5 z + 8 w = 30 \
2 x + 3 y + 6 z + 9 w = 15
\end{array}$$
A) $\{ ( - 6 w + 3,9 w + 7 , - 4 w - 2 , w ) \}$
B) $\{ ( 5 w + 11 , - 3 w - 7 , - 3 w + 4 , w ) \}$
C) $\{ ( - 3,16 , - 6,1 ) \}$
D) $\varnothing$

Quizplus · Accepted Answer

Applying Gaussian elimination to the given system of equations will reduce it to a form where one variable, in this case $w$, can be chosen freely (a parameter), and the other variables $x$, $y$, and $z$ are expressed in terms of $w$. This results in an infinite number of solutions, represented by the set in option A, where each variable is a linear function of $w$.