Question

Examine whether the following system of equations is consistent or inconsistent. If consistent, find the complete solution
2x + 3y + z = 9
x + 2y + 3z = 6
3x + y + 2z = 8

Answer 1

Answer:

Given equations are consistent and has unique solution ,because rank is equal to number of variables.

$x=\frac{35}{18} ,y=\frac{29}{18},z=\frac{5}{18}$

Step-by-step explanation:

To check the consistency of the given system of linear equations,we must find out the rank of augmented matrix and coefficient matrix,if both are equal than only given system of linear equations are consistent,otherwise inconsistent.

Augmented matrix:

$A=\left[\begin{array}{cccc}2&3&1&9\\1&2&3&6\\3&1&2&8\end{array}\right]\\\\\\R_{2} -> R_{2}-\frac{1}{2}R_{1}\\\\\left[\begin{array}{cccc}2&3&1&9\\0&\frac{1}{2}&\frac{5}{2}&\frac{3}{2}\\\\3&1&2&8\end{array}\right]\\\\\\R_{3} -> R_{3}-\frac{3}{2}R_{1}\\\\\left[\begin{array}{cccc}2&3&1&9\\0&\frac{1}{2}&\frac{5}{2}&\frac{3}{2}\\\\0&\frac{-7}{2}&\frac{1}{2}&\frac{-11}{2}\end{array}\right]\\\\\\R_{3} -> R_{3}+7R_{2}\\\\\left[\begin{array}{cccc}2&3&1&9\\0&\frac{1}{2}&\frac{5}{2}&\frac{3}{2}\\\\0&0&18&5\end{array}\right]$

$Rank\left[\begin{array}{cccc}2&3&1&9\\1&2&3&6\\3&1&2&8\end{array}\right]=Rank\left[\begin{array}{cccc}2&3&1&9\\0&\frac{1}{2}&\frac{5}{2}&\frac{3}{2}\\\\0&0&18&5\end{array}\right]=3$

it is clear from augmented matrix,thus rank of cofficient matrix is 3.

So,equations are consistent and has unique solution ,because rank is equal to number of variables.

Cramer's Rule:

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

$\triangle=\left|\begin{array}{ccc}2&3&1\\1&2&3\\3&1&2\end{array}\right|=18$

$\triangle_{1}=\left|\begin{array}{ccc}9&3&1\\6&2&3\\8&1&2\end{array}\right|=35$

$\triangle_{2}=\left|\begin{array}{ccc}2&9&1\\1&6&3\\3&8&2\end{array}\right|=29$

$\triangle_{3}=\left|\begin{array}{ccc}2&3&9\\1&2&6\\3&1&8\end{array}\right|=5$

$x=\frac{35}{18} ,y=\frac{29}{18},z=\frac{5}{18}$