Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices.
$\begin{array} { r }
\left\{ \begin{aligned}
x + 5 y - 5 z & = 42 \
y - 9 z & = 48 \
z & = - 5
\end{aligned} \right. \
\left\{ \begin{aligned}
x + 8 y - 3 z & = 31 \
y + 5 z & = - 22 \
z & = - 5
\end{aligned} \right.
\end{array}$
A) $x = 2 , y = 3 , z = - 5$
B) $x = - 2 , y = - 3 , z = - 5$
C) $x = - 3 , y = - 5 , z = 2$
D) $x = 3 , y = - 5 , z = 2$
E) The systems yield different solutions.

Question

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices.
$\begin{array} { r } 
\left\{ \begin{aligned}
x + 5 y - 5 z & = 42 \
y - 9 z & = 48 \
z & = - 5
\end{aligned} \right. \
\left\{ \begin{aligned}
x + 8 y - 3 z & = 31 \
y + 5 z & = - 22 \
z & = - 5
\end{aligned} \right.
\end{array}$
A) $x = 2 , y = 3 , z = - 5$
B) $x = - 2 , y = - 3 , z = - 5$
C) $x = - 3 , y = - 5 , z = 2$
D) $x = 3 , y = - 5 , z = 2$
E) The systems yield different solutions.

Quizplus · Accepted Answer

Given that both systems have $z = -5$ as a solution, we can substitute $z = -5$ into the other equations of each system to check for consistency. For the first system, substituting $z = -5$ into the second equation gives $y - 9(-5) = 48$, which simplifies to $y + 45 = 48$, leading to $y = 3$. Substituting $y = 3$ and $z = -5$ into the first equation gives $x + 5(3) - 5(-5) = 42$, which simplifies to $x + 15 + 25 = 42$, leading to $x = 2$. For the second system, substituting $z = -5$ into the second equation gives $y + 5(-5) = -22$, which simplifies to $y - 25 = -22$, leading to $y = 3$. However, substituting $y = 3$ and $z = -5$ into the first equation of the second system gives $x + 8(3) - 3(-5) = 31$, which simplifies to $x + 24 + 15 = 31$, leading to $x = -8$, not matching the solution from the first system. Therefore, the systems yield different solutions.

Determine Whether the Two Systems of Linear Equations Yield the Same