Find the solution or solutions, if any exist, to the system.
-$$\left\{ \begin{array} { c }
x + 4 y - 2 z = - 5 \
3 x - 6 y + 6 z = 7 \
5 x + 2 y + 2 z = - 3
\end{array} \right.$$
A) infinitely many solutions of the form $x = - \frac { 1 } { 9 } - \left( \frac { 2 } { 3 } \right) z , y = - \frac { 11 } { 9 } + \left( \frac { 2 } { 3 } \right) z , z = z$
B) $x = 0 , y = 0 , z = 1$
C) infinitely many solutions of the form $x = 8 - \frac { 2 } { 9 } z , y = 1 - \frac { 17 } { 18 } z , z = z$
D) inconsistent system, no solution

Question

Find the solution or solutions, if any exist, to the system.
-$$\left\{ \begin{array} { c } 
x + 4 y - 2 z = - 5 \
3 x - 6 y + 6 z = 7 \
5 x + 2 y + 2 z = - 3
\end{array} \right.$$
A) infinitely many solutions of the form $x = - \frac { 1 } { 9 } - \left( \frac { 2 } { 3 } \right) z , y = - \frac { 11 } { 9 } + \left( \frac { 2 } { 3 } \right) z , z = z$
B) $x = 0 , y = 0 , z = 1$
C) infinitely many solutions of the form $x = 8 - \frac { 2 } { 9 } z , y = 1 - \frac { 17 } { 18 } z , z = z$
D) inconsistent system, no solution

Quizplus · Accepted Answer

To solve the system of equations, we can use either substitution or elimination method. By attempting to eliminate one variable at a time, we can find that the system does not contradict and is dependent, leading to infinitely many solutions. The coefficients and constants when manipulated correctly will lead to the expressions given in option A for $x$ and $y$ in terms of $z$, indicating that for any value of $z$, we can find corresponding values of $x$ and $y$, hence an infinite number of solutions.

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