Solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent.
-$$\begin{aligned}
x - 2 z & = 10 \
3 y + 6 z & = - 27 \
2 x - 2 y & = 6
\end{aligned}$$
A) $\{ ( 10,0,0 ) \}$
B) Infinitely many solutions; dependent
C) $\{ ( 2 , - 1 , - 4 ) \}$
D) No solution; inconsistent

Question

Quizplus · Accepted Answer

The system has a unique solution, which can be found by solving the equations simultaneously. By manipulating and substituting the equations, we can find the values of $x$, $y$, and $z$ that satisfy all three equations, leading to the solution set $\{ ( 2 , - 1 , - 4 ) \}$.

Solve the System x−2z=103y+6z=−272x−2y=6\begin{aligned}x - 2 z & = 10 \\3 y + 6 z & = - 27 \\2 x - 2 y & = 6\end{aligned}x−2z3y+6z2x−2y​=10=−27=6​

Solve the System $\begin{aligned}x - 2 z & = 10 \\3 y + 6 z & = - 27 \\2 x - 2 y & = 6\end{aligned}$