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  • Examine the consistency of the system of equations x + 3y = 5, 2x + 6y = 8.

Q : 3         Examine the consistency of the system of equations.

                 \small x+3y=5

                \small 2x+6y=8

Answers (1)

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We have given the system of equations:

                                      \small x+3y=5

                                    \small 2x+6y=8

The given system of equations can be written in the form of the matrix; AX =B

where A= \begin{bmatrix} 1 &3 \\ 2&6 \end{bmatrix},  X= \begin{bmatrix} x\\y \end{bmatrix}  and B = \begin{bmatrix} 5\\8 \end{bmatrix}.

So, we want to check for the consistency of the equations;

|A| = 1(6) -2(3) = 0

Here A is singular matrix therefore now we will check whether the (adjA)B is zero or non-zero.

adjA= \begin{bmatrix} 6 &-3 \\ -2& 1 \end{bmatrix}

So, (adjA)B= \begin{bmatrix} 6 &-3 \\ -2& 1 \end{bmatrix}\begin{bmatrix} 5\\8 \end{bmatrix} = \begin{bmatrix} 30-24\\-10+8 \end{bmatrix}=\begin{bmatrix} 6\\-2 \end{bmatrix} \neq 0

As, (adjA)B \neq 0 , the solution of the given system of equations does not exist.

Hence, the given system of equations is inconsistent.

Posted by

Divya Prakash Singh

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