Respuesta :
Answer with explanation:
The given system of equation are
3x + 5y - 2w = -13
2x + 7z - w = -1
4y + 3z + 3w = 1
-x + 2y + 4z = -5
Writing the system of equation in terms of Augmented Matrix
[tex]\left[\begin{array}{ccccc}3&5&0&-2&-13\\2&0&7&-1&-1\\0&4&3&3&1\\-1&2&4&0&-5\end{array}\right]\\\\R_{3} \leftrightarrow R_{4}\\\\\left[\begin{array}{ccccc}3&5&0&-2&-13\\2&0&7&-1&-1\\-1&2&4&0&-5\\0&4&3&3&1\end{array}\right]\\\\R_{3} \rightarrow 2R_{3}+R_{2}\\\\\left[\begin{array}{ccccc}3&5&0&-2&-13\\2&0&7&-1&-1\\0&4&15&-1&-11\\0&4&3&3&1\end{array}\right]\\\\R_{4}\rightarrow R_{4}-R_{3}\\\\\left[\begin{array}{ccccc}3&5&0&-2&-13\\2&0&7&-1&-1\\0&4&15&-1&-11\\0&0&-12&4&12\end{array}\right][/tex]
[tex]R_{2}\rightarrow 3R_{2}-2R_{1}\\\\\left[\begin{array}{ccccc}3&5&0&-2&-13\\0&-10&21&1&23\\0&4&15&-1&-11\\0&0&-12&4&12\end{array}\right]\\\\R_{3}\rightarrow 5R_{3}+2R_{2}\\\\\left[\begin{array}{ccccc}3&5&0&-2&-13\\0&-10&21&1&23\\0&0&117&-3&-9\\0&0&-12&4&12\end{array}\right]\\\\ R_{4}\rightarrow 3R_{4}+4R_{3}\\\\\left[\begin{array}{ccccc}3&5&0&-2&-13\\0&-10&21&1&23\\0&0&117&-3&-9\\0&0&432&0&0\end{array}\right][/tex]
→432 z=0
z=0
⇒117 z-3 w=-9
-3 w=-9
Dividing both sides by -3
w=3
⇒-10 y+21z+w=23
-10 y+0+3=23
-10 y=23-3
-10 y= 20
y=-2
⇒3 x+5 y-2w=-13
3 x+5 ×(-2)-2 ×3= -13
3 x-10-6= -13
3 x=16-13
3 x=3
x=1
Option B. {(1, -2, 0, 3)}
