Question

Examine whether the following system of equations is consistent or inconsistent. If consistent, find the complete solutionx + y + z = 92x + 5y + 7z = 522x + y - z = 0

Answer 1

Answer:

Given sysytem of linear equations are consistent and have unique solution.

$x=0,y=0,z=0$

Step-by-step explanation:

Analysis of equation: Since here  

$A=\left[\begin{array}{ccc}1&1&1\\92&5&7\\522&1&-1\end{array}\right] \\\\\\X=\left[\begin{array}{ccc}x\\y\\z\end{array}\right] \\\\\\B=\left[\begin{array}{ccc}0\\0\\0\end{array}\right]$

$Rank\left[\begin{array}{cccc}1&1&1&0\\92&5&7&0\\522&1&-1&0\end{array}\right]=3\\\\Rank\left[\begin{array}{ccc}1&1&1\\92&5&7\\522&1&-1\end{array}\right]=3$

SInce rank of both matrix are equal and equal to the number of variables,thus given sysytem of linear equations are consistent and have unique solution.

Solution by Cramer's Rule:

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

$\triangle=\left|\begin{array}{ccc}1&1&1\\92&5&7\\522&1&-1\end{array}\right|=1216$

$\triangle_{1}=\left|\begin{array}{ccc}0&1&1\\0&5&7\\0&1&-1\end{array}\right|=0$

$\triangle_{2}=\left|\begin{array}{ccc}1&0&1\\92&0&7\\522&0&-1\end{array}\right|=0$

$\triangle_{3}=\left|\begin{array}{ccc}1&1&0\\92&5&0\\522&1&0\end{array}\right|=0$

$x=\frac{0}{1216} ,y=\frac{0}{1216},z=\frac{0}{1216}$

$x=0,y=0,z=0$