Find the solution or solutions, if any exist, to the system.
-$$\left\{ \begin{array} { c }
2 x - 6 z - 6 w = - 4 \
x + y + z + 3 w = 2 \
4 x + 2 y - 4 z - 4 w = 0 \
6 x + 4 y - 2 z + 2 w = 4
\end{array} \right.$$
A) infinitely many solutions of the form $x = 3 z - 2 , y = - 4 z + 4 , z = z , w = 0$
B) inconsistent system, no solution
C) $x = - 2 , y = 4 , z = 0 , w = 0$
D) infinitely many solutions of the form $x = - 3 z - 2 , y = 4 z + 4 , z = 0 , w = w$

Question

Find the solution or solutions, if any exist, to the system.
-$$\left\{ \begin{array} { c } 
2 x - 6 z - 6 w = - 4 \
x + y + z + 3 w = 2 \
4 x + 2 y - 4 z - 4 w = 0 \
6 x + 4 y - 2 z + 2 w = 4
\end{array} \right.$$
A) infinitely many solutions of the form $x = 3 z - 2 , y = - 4 z + 4 , z = z , w = 0$
B) inconsistent system, no solution
C) $x = - 2 , y = 4 , z = 0 , w = 0$
D) infinitely many solutions of the form $x = - 3 z - 2 , y = 4 z + 4 , z = 0 , w = w$

Quizplus · Accepted Answer

This system of equations has infinitely many solutions. By manipulating the equations, you can express some variables in terms of others, indicating that there are free variables which lead to an infinite number of solutions. The correct form of these solutions is given in option A, where $x$, $y$, and $w$ are expressed in terms of $z$, and $z$ is free to be any value.

Find the Solution or Solutions, If Any Exist, to the System