$\left\{ \begin{aligned}
6 y + w & = - 25 \
x + y + 4 z - w & = 14 \
6 x + z + 4 w & = 6 \
x + y + 4 z & = 13
\end{aligned} \right.$
A) x=1, y=-4, z=4, w=-1
B) x=6, y=-24, z=16, w=-6
C) x=1, y=-24, z=8, w=-2
D) x=1, y=-4, z=16, w=-1

Question

$\left\{ \begin{aligned}
6 y + w & = - 25 \
x + y + 4 z - w & = 14 \
6 x + z + 4 w & = 6 \
x + y + 4 z & = 13
\end{aligned} \right.$
A)  x=1, y=-4, z=4, w=-1 
B)  x=6, y=-24, z=16, w=-6 
C)  x=1, y=-24, z=8, w=-2 
D)  x=1, y=-4, z=16, w=-1

Quizplus · Accepted Answer

The system of equations can be solved by various methods such as substitution, elimination, or matrix inversion. By comparing the third and fourth equations, it's clear that $w = -1$ since subtracting the fourth equation from the third gives $6x + z + 4w - (x + y + 4z) = 6 - 13$, which simplifies to $5x - 3z + 4w = -7$, and knowing $w = -1$ helps in further simplifications. Substituting $w = -1$ into the first equation gives $6y - 1 = -25$, leading to $y = -4$. The fourth equation then simplifies to $x - 4 + 4z = 13$, which, along with other equations, helps in determining that $x = 1$ and $z = 4$. This matches option A.

{6y+w=−25x+y+4z−w=146x+z+4w=6x+y+4z=13\left\{ \begin{aligned}6 y + w & = - 25 \\x + y + 4 z - w & = 14 \\6 x + z + 4 w & = 6 \\x + y + 4 z & = 13\end{aligned} \right.⎩⎨⎧​6y+wx+y+4z−w6x+z+4wx+y+4z​=−25=14=6=13​

$\left\{ \begin{aligned}6 y + w & = - 25 \\x + y + 4 z - w & = 14 \\6 x + z + 4 w & = 6 \\x + y + 4 z & = 13\end{aligned} \right.$