Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. $\left\{ \begin{aligned}
x - 7 y - 2 z & = 43 \
y + 2 z & = - 14 \
z & = - 4
\end{aligned} \right.$ $\left\{ \begin{aligned}
x + 2 y + 3 z & = - 27 \
y - 2 z & = 2 \
z & = - 4
\end{aligned} \right.$
A)x = -7, y = -6, z = -4
B)x = 7, y = 6, z = -4
C)x = 6, y = -4, z = -7
D)x = -6, y = -4, z = -7
E)The systems yield different solutions.

Question

Quizplus · Accepted Answer

The third equation in both systems, $z = -4$, is the same, indicating that at least part of the solution will be shared. However, substituting $z = -4$ into the second equations of both systems yields different equations for $y$, indicating that the systems will not have the same solution for $x$ and $y$. Specifically, substituting $z = -4$ into the second equation of the first system gives $y + 2(-4) = -14$, which simplifies to $y = -6$. For the second system, substituting $z = -4$ into the second equation gives $y - 2(-4) = 2$, which simplifies to $y = -6$. Despite the same value for $y$, when we look at the structure and coefficients of the equations in both systems, they are different, leading to potentially different values for $x$. Therefore, without further calculation, it's clear that the overall solutions for $x$, $y$, and $z$ might not match across the two systems, indicating that they yield different solutions.

Determine Whether the Two Systems of Linear Equations Yield the Same