Solve the system of linear equations.
$$\left\{ \begin{aligned}
x + y + z + w & = 11 \
- 2 x - y - 3 z + 4 w & = - 5 \
- 2 x - 5 z - w & = - 23 \
3 x + 5 y + z - 2 w & = 16
\end{aligned} \right.$$
A) $( 5,1,3,2 )$
B) $( 5,2,1,3 )$
C) $( 3,1,2,5 )$
D) $( 5,1,2,3 )$
E) inconsistent

Question

Quizplus · Accepted Answer

We can solve this system using row reduction on the augmented coefficient matrix: $$\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 11 \ -2 & -1 & -3 & 4 & -5\ -2 & 0 & -5 & -1 & -23 \ 3 & 5 & 1 & -2 & 16 \end{array}ight] $$ The first step is to subtract two times the first row from the second row and two times the first row from the third row: $$\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 11 \ 0 & -3 & -5 & 6 & 17\ 0 & -2 & -7 & -3 &-1 \ 3 & 5 & 1 & -2 & 16 \end{array}ight] $$ The next step is to add two times the second row to the fourth row: $$\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 11 \ 0 & -3 & -5 & 6 & 17\ 0 & -2 & -7 & -3 &-1 \ 0 & -1 & -9 & 10 & 50 \end{array}ight] $$ Now we can add three times the third row to the second row: $$\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 11 \ 0 & 1 & -28 & 3 & -4\ 0 & -2 & -7 & -3 &-1 \ 0 & -1 & -9 & 10 & 50 \end{array}ight] $$ Next, subtract one times the second row from the fourth row: $$\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 11 \ 0 & 1 & -28 & 3 & -4\ 0 & -2 & -7 & -3 &-1 \ 0 & 0 & 19 & 7 & 46 \end{array}ight] $$ Finally, we can subtract two times the third row from the fourth row: $$\left[\begin{array}{cccc|c} 1 & 1 & 1 & 1 & 11 \ 0 & 1 & -28 & 3 & -4\ 0 & -2 & -7 & -3 &-1 \ 0 & 0 & 5 & 13 & 48 \end{array}ight] $$ Thus, $w=3$, $z=2$, $y = 1$, and $x = 5$. The answer is $\boxed{	extbf{(D)}}$.

Solve the System of Linear Equations A) (5,1,3,2)( 5,1,3,2 )(5,1,3,2) B) (5,2,1,3)( 5,2,1,3 )(5,2,1,3)

Solve the System of Linear Equations A) $( 5,1,3,2 )$ B) $( 5,2,1,3 )$