Solve the system of equations below by the Gaussian elimination method: $3 t + 6 u + v - 8 w = - 31$ $t + 3 u - 2 v + 7 w = - 21$ $- 7 t - 2 u + 9 v + 6 w = 23$
A) $\left( \frac { 333 } { 59 } c + 1 , - \frac { 120 } { 59 } c - 6 , \frac { 193 } { 59 } c + 2 , c \right)$
B) $\left( \frac { 333 } { 59 } c + 1 , \frac { 193 } { 59 } c + 2 , - \frac { 120 } { 59 } c - 6 , c \right)$
C) $\left( \frac { 193 } { 59 } c + 2 , \frac { 333 } { 59 } c - 6 , - \frac { 120 } { 59 } c + 1 , c \right)$
D) $\left( - \frac { 120 } { 59 } c - 6 , \frac { 193 } { 59 } c + 1 , \frac { 333 } { 59 } c + 2 , c \right)$
E)no solution

Question

Quizplus · Accepted Answer

The system of equations given is underdetermined, meaning it has more variables than equations, which typically results in an infinite number of solutions. By applying the Gaussian elimination method, we can reduce the system to row echelon form, and then to reduced row echelon form, to find the relationships between the variables. The correct solution set indicates that one of the variables (in this case, $c$) can be chosen freely, and the other variables are then determined in terms of $c$. The coefficients and constants in choice A correctly represent the relationships between the variables after applying Gaussian elimination, indicating that the system has an infinite number of solutions parameterized by $c$.

Solve the System of Equations Below by the Gaussian Elimination 3t+6u+v−8w=−313 t + 6 u + v - 8 w = - 313t+6u+v−8w=−31

Solve the System of Equations Below by the Gaussian Elimination $3 t + 6 u + v - 8 w = - 31$