Solve the system.
$\begin{array} { c }
x - y + 5 z = 27 \
2 x + z = 5 \
x + 4 y + z = - 3
\end{array}$
A) $\{ ( 0 , - 2,5 ) \}$
B) $\{ ( 5 , - 2,0 ) \}$
C) $\{ ( 5,0 , - 2 ) \}$
D) $\varnothing$

Question

Solve the system.
$\begin{array} { c } 
x - y + 5 z = 27 \
2 x + z = 5 \
x + 4 y + z = - 3
\end{array}$
A) $\{ ( 0 , - 2,5 ) \}$
B) $\{ ( 5 , - 2,0 ) \}$
C) $\{ ( 5,0 , - 2 ) \}$
D) $\varnothing$

Quizplus · Accepted Answer

Explanation:Solve the system by elimination.$\begin{array} { c } x - y + 5 z = 27 \2 x + z = 5 \x + 4 y + z = - 3\end{array}$Subtract the second equation from the first equation.$\begin{array} { c } x - y + 5 z = 27 \- x - z = - 5 \x + 4 y + z = - 3\end{array}$Add the second equation to the third equation.$\begin{array} { c } x - y + 5 z = 27 \- x - z = - 5 \4 y = - 8\end{array}$Solve the third equation for y.$\begin{array} { c } x - y + 5 z = 27 \- x - z = - 5 \y = - 2\end{array}$Substitute the value of y into the second equation.$\begin{array} { c } x - y + 5 z = 27 \2 x + z = 5 \- 2 = - 2\end{array}$The third equation is an identity, so it does not provide any new information.Substitute the value of y into the first equation.$\begin{array} { c } x - (- 2) + 5 z = 27 \2 x + z = 5 \- 2 = - 2\end{array}$Solve the first equation for x.$\begin{array} { c } x + 2 + 5 z = 27 \2 x + z = 5 \- 2 = - 2\end{array}$$\begin{array} { c } x + 5 z = 25 \2 x + z = 5 \- 2 = - 2\end{array}$$\begin{array} { c } x = 5 - 5 z \2 ( 5 - 5 z ) + z = 5 \- 2 = - 2\end{array}$$\begin{array} { c } x = 5 - 5 z \10 - 10 z + z = 5 \- 2 = - 2\end{array}$$\begin{array} { c } x = 5 - 5 z \9 - 10 z = 5 \- 2 = - 2\end{array}$$\begin{array} { c } x = 5 - 5 z \- 10 z = - 4 \- 2 = - 2\end{array}$$\begin{array} { c } x = 5 - 5 z \z = 2/5 \- 2 = - 2\end{array}$$\begin{array} { c } x = 5 - 5 ( 2/5 ) \z = 2/5 \- 2 = - 2\end{array}$$\begin{array} { c } x = 0 \z = 2/5 \- 2 = - 2\end{array}$Therefore, the solution to the system is $( 0 , - 2,2/5 )$.

Solve the System x−y+5z=272x+z=5x+4y+z=−3\begin{array} { c } x - y + 5 z = 27 \\2 x + z = 5 \\x + 4 y + z = - 3\end{array}x−y+5z=272x+z=5x+4y+z=−3​

Solve the System $\begin{array} { c } x - y + 5 z = 27 \\2 x + z = 5 \\x + 4 y + z = - 3\end{array}$