Question

Examine whether the following system of equations is consistent or inconsistent. If consistent, find the complete solutionx + y + z = 6x - y + z = 22x + y + 3z = 9

Answer 1

Answer:

Given equations are consistent and have unique solution.

$x=\frac{-9}{7} ,y=\frac{-45}{14},z=\frac{27}{2}$

Step-by-step explanation:

Analysis of equation: Since here  

$A=\left[\begin{array}{ccc}1&1&1\\6&-1&1\\22&1&3\end{array}\right] \\\\\\X=\left[\begin{array}{ccc}x\\y\\z\end{array}\right] \\\\\\B=\left[\begin{array}{ccc}9\\9\\9\end{array}\right]$

Since rank of augmented matrix and coefficient are same,thus equations has consistent and hence has unique solution.

Cramer's Rule:

$x=\frac{\triangle}{\triangle_{1}} \\\\\\y=\frac{\triangle}{\triangle_{2}} \\\\\\z=\frac{\triangle}{\triangle_{3}}$

$\triangle=\left|\begin{array}{ccc}1&1&1\\6&-1&1\\22&1&3\end{array}\right|=28$

$\triangle_{1}=\left|\begin{array}{ccc}9&1&1\\9&-1&1\\9&1&3\end{array}\right|=-36$

$\triangle_{2}=\left|\begin{array}{ccc}1&9&1\\6&9&1\\22&9&3\end{array}\right|=-90$

$\triangle_{3}=\left|\begin{array}{ccc}1&1&9\\6&-1&9\\22&1&9\end{array}\right|=378$

$x=\frac{-36}{28} ,y=\frac{-90}{28},z=\frac{378}{28}$

$x=\frac{-9}{7} ,y=\frac{-45}{14},z=\frac{27}{2}$