Question

Examine whether the following system of equations is consistent or inconsistent. If consistent, find the complete solutionx + y + z = 6x + 2y + 3z = 10x + 2y + 4z = 1

Answer 1

Answer:

The rank of the augmented matrix is not equal to the rank of the coefficient matrix of the system. Thus system is inconsistent (has no solution at all)

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:

$A=\left[\begin{array}{cccc}1&1&1&1\\6&2&3&1\\10&2&4&1\end{array}\right]\\\\R_{2} -> R_{2}-6R_{1}\\\\\left[\begin{array}{cccc}1&1&1&1\\0&-4&-3&-5\\10&2&4&1\end{array}\right]\\\\R_{3} -> R_{3}-10R_{1}\\\\\left[\begin{array}{cccc}1&1&1&1\\0&-4&-3&-5\\0&-8&-6&-9\end{array}\right]\\\\R_{3} -> R_{3}-2R_{2}\\\\\left[\begin{array}{cccc}1&1&1&1\\0&-4&-3&-5\\0&0&0&1\end{array}\right]$

$Rank\left[\begin{array}{cccc}1&1&1&1\\6&2&3&1\\10&2&4&1\end{array}\right]=Rank\left[\begin{array}{cccc}1&1&1&1\\0&-4&-3&-5\\0&0&0&1\end{array}\right]=3$

hence rank of coefficient matrix

$Rank\left[\begin{array}{ccc}1&1&1\\6&2&3\\10&2&4\end{array}\right]=Rank\left[\begin{array}{ccc}1&1&1\\0&-4&-3\\0&0&0\end{array}\right]=2$

The rank of the augmented matrix is not equal to the rank of the coefficient matrix of the system. Thus system is inconsistent (has no solution at all)