Examine the consistency of the following system of equations. $x + 3y = 5$ and $2x + 6y = 8$

Inconsistent

The correct answer is Option (3) → Inconsistent ##

Given set of equations is: $x + 3y = 5$ and $2x + 6y = 8$

This set of equation can be written in the form of matrix as $AX = B$.

$\begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 8 \end{bmatrix}$

where, $A = \begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 5 \\ 8 \end{bmatrix}$

$|A| = \begin{vmatrix} 1 & 3 \\ 2 & 6 \end{vmatrix} = 0$

$\text{adj. } A = \begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}$

And $(\text{adj. } A)B = \begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} 5 \\ 8 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix} \neq 0$

Hence, the given equations form inconsistent system and has no solution.