Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices.
$$\begin{aligned}
\left\{ \begin{aligned}
x + 9 y - 2 z & = 41 \
y - 7 z & = 46 \
z & = - 6
\end{aligned} \right.\
\left\{ \begin{aligned}
x + 4 y + 2 z & = - 3 \
y + 5 z & = - 26 \
z & = - 6
\end{aligned} \right.
\end{aligned}$$
A) $x = - 7 , y = 4 , z = - 6$
B) $x = 7 , y = - 4 , z = - 6$
C) $x = - 4 , y = - 6 , z = - 7$
D) $x = 4 , y = - 6 , z = - 7$
E) The systems yield different solutions.

Question

Quizplus · Accepted Answer

The augmented matrices for the two systems are$$\left[ \begin{array}{rrr|r} 1 & 9 & -2 & 41 \\ 0 & 1 & -7 & 46 \\ 0 & 0 & 1 & -6 \end{array} \right] \quad \text{ and } \quad \left[ \begin{array}{rrr|r} 1 & 4 & 2 & -3 \\ 0 & 1 & 5 & -26 \\ 0 & 0 & 1 & -6 \end{array} \right]$$Reducing both matrices to row echelon form gives$$\left[ \begin{array}{rrr|r} 1 & 0 & -11 & 7 \\ 0 & 1 & -7 & 46 \\ 0 & 0 & 1 & -6 \end{array} \right] \quad \text{ and } \quad \left[ \begin{array}{rrr|r} 1 & 0 & -11 & 7 \\ 0 & 1 & -7 & 46 \\ 0 & 0 & 1 & -6 \end{array} \right]$$Since the row echelon forms are the same, the two systems yield the same solutions. Solving the system$$\begin{aligned}x - 11 z & = 7 \y - 7 z & = 46 \z & = -6\end{aligned}$$gives $x = -7 , y = 4 , z = - 6$.

Determine Whether the Two Systems of Linear Equations Yield the Same