Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices.
$\left\{ \begin{aligned} x + y - 9 z & = - 67 \ y + 5 z & = 31 \ z & = 7 \end{aligned} \right.$
$\left\{ \begin{aligned} x - 9 y + 6 z & = 76 \ y - 9 z & = - 67 \ z & = 7 \end{aligned} \right.$
A) $x = 0 , y = - 4 , z = 7$
B) $x = 0 , y = 4 , z = 7$
C) $x = 4 , y = 7 , z = 0$
D) $x = - 4 , y = 7 , z = 0$
E) The systems yield different solutions.

Question

Quizplus · Accepted Answer

The third equation in both systems directly gives $z = 7$. Substituting $z = 7$ into the second equation of the first system gives $y + 5(7) = 31$, which simplifies to $y + 35 = 31$, or $y = -4$. Substituting $y = -4$ and $z = 7$ into the first equation of the first system gives $x - 4 - 9(7) = -67$, which simplifies to $x - 4 - 63 = -67$, or $x = 0$. Thus, for the first system, $x = 0$, $y = -4$, and $z = 7$.For the second system, substituting $z = 7$ into the second equation gives $y - 9(7) = -67$, which simplifies to $y - 63 = -67$, or $y = -4$, matching the $y$ value from the first system. However, substituting $y = -4$ and $z = 7$ into the first equation of the second system gives $x - 9(-4) + 6(7) = 76$, which simplifies to $x + 36 + 42 = 76$, or $x = -2$, which does not match the $x$ value from the first system.Given this discrepancy in the $x$ values derived from the two systems, it's clear that the two systems do not yield the same solutions, despite initially appearing to do so based on the $y$ and $z$ values. Therefore, the correct choice is that the systems yield different solutions.

Determine Whether the Two Systems of Linear Equations Yield the Same