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} { l }
x - y + z - w = 10 \
\quad - 2 x + 3 y + 5 w = - 28 \
x + 2 y + 8 z + 3 w = - 10 \
x - 4 y - 6 z - 5 w = 30 \

A){ ( - 17 w - 10 , - 13 w - 16,5 w + 4 , w ) }
B) { ( 3 w - 2 , - 8 w + 3,4 w + 9 , w ) }
C) { ( 24,10 , - 6 , - 2 ) }
D) $ \varnothing $

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} { l } 
x - y + z - w = 10 \
\quad - 2 x + 3 y + 5 w = - 28 \
x + 2 y + 8 z + 3 w = - 10 \
x - 4 y - 6 z - 5 w = 30 \

A){ ( - 17 w - 10 , - 13 w - 16,5 w + 4 , w ) }  
B) { ( 3 w - 2 , - 8 w + 3,4 w + 9 , w ) } 
C) { ( 24,10 , - 6 , - 2 ) } 
D) $ \varnothing $

Quizplus · Accepted Answer

The system of equations does not have a unique solution, but it is consistent with infinitely many solutions. By applying Gaussian elimination, we can reduce the system to a form that reveals the dependency of some variables on others, typically leading to a parametric solution where one or more variables are free to take any value, and the others are expressed in terms of these free variables. The correct solution set is given by option A, indicating a dependency of the variables on $w$, which acts as a parameter. This reflects the nature of the system having infinitely many solutions, where each solution can be expressed in terms of $w$.

Apply Gaussian Elimination to Systems Without Unique Solutions Use Gaussian \[\Begin{array} { L } X - Y + Z

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian \[\Begin{array} { L }
X - Y + Z