Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices. $$\left\{ \begin{aligned}
x + 2 y - 4 z & = - 8 \
y - 7 z & = 4 \
z & = - 1
\end{aligned} \right.$$ $$\left\{ \begin{aligned}
x - y - 5 z & = 21 \
y - 6 z & = 3 \
z & = - 1
\end{aligned} \right.$$
A)x = 6, y = 3, z = -1
B)x = -6, y = -3, z = -1
C)x = 3, y = -1, z = -6
D)The systems yield different solutions.
E)x = -3, y = -1, z = -6

Question

Quizplus · Accepted Answer

The third equation in both systems directly gives $z = -1$. Substituting $z = -1$ into the second equations of both systems gives different values for $y$: $y - 7(-1) = 4$ leads to $y = -3$ for the first system, and $y - 6(-1) = 3$ leads to $y = -3$ for the second system, which actually matches. However, substituting $y = -3$ and $z = -1$ into the first equations of both systems gives different values for $x$, which indicates the systems yield different solutions. The mistake in my explanation shows that the initial assessment was incorrect. Let's correct this:For the first system, substituting $z = -1$ into the second equation gives $y - 7(-1) = 4$, which simplifies to $y = -3$. Substituting $y = -3$ and $z = -1$ into the first equation gives $x + 2(-3) - 4(-1) = -8$, which simplifies to $x = -6$.For the second system, substituting $z = -1$ into the second equation gives $y - 6(-1) = 3$, which also simplifies to $y = -3$. Substituting $y = -3$ and $z = -1$ into the first equation gives $x - (-3) - 5(-1) = 21$, which simplifies to $x = -6$.Therefore, both systems yield the same solution: $x = -6$, $y = -3$, $z = -1$. The correct choice is:B

Determine Whether the Two Systems of Linear Equations Yield the Same