Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{aligned}
2 x + y + 2 z - 4 w & = 10 \
x + 3 y + 2 z - 11 w & = 17 \
3 x + y + 7 z - 21 w & = 0
\end{aligned}$$
A) $\{ ( - 3 w + 5,2 w + 6,4 w - 3 , w ) \}$
B) $\{ ( 3 w + 5,6 w + 6 , - 4 w - 3 , w ) \}$
C) $\{ ( w + 5,8 w + 4 , - 3 w - 2 , w ) \}$
D) $\{ ( w - 5,8 w - 4 , - 3 w + 2 , w ) \}$

Question

Quizplus · Accepted Answer

The system has more variables (4) than equations (3), suggesting at least one variable is free. By applying Gaussian elimination, we can reduce the system to row-echelon form and solve for the variables in terms of the free variable(s). The correct solution set, $\{ ( - 3 w + 5,2 w + 6,4 w - 3 , w ) \}$, indicates that $w$ is chosen as the free variable, and the other variables ($x, y, z$) are expressed in terms of $w$. This solution set correctly represents the relationships among the variables after applying Gaussian elimination to the given system of equations.

Apply Gaussian Elimination to Systems with More Variables Than Equations 2x+y+2z−4w=10x+3y+2z−11w=173x+y+7z−21w=0\begin{aligned}2 x + y + 2 z - 4 w & = 10 \\x + 3 y + 2 z - 11 w & = 17 \\3 x + y + 7 z - 21 w & = 0\end{aligned}2x+y+2z−4wx+3y+2z−11w3x+y+7z−21w​=10=17=0​

Apply Gaussian Elimination to Systems with More Variables Than Equations $\begin{aligned}2 x + y + 2 z - 4 w & = 10 \\x + 3 y + 2 z - 11 w & = 17 \\3 x + y + 7 z - 21 w & = 0\end{aligned}$