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# Examine the consistency of the system of equations

Question:

Examine the consistency of the system of equations

$x+2 y=2$

$2 x+3 y=3$

Solution:

The given system of equations is:

$x+2 y=2$

$2 x+3 y=3$

The given system of equations can be written in the form of AX = B, where

$A=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right], X=\left[\begin{array}{l}x \\ y\end{array}\right]$ and $B=\left[\begin{array}{l}2 \\ 3\end{array}\right]$

Now,

$|A|=1(3)-2(2)=3-4=-1 \neq 0$

$\therefore A$ is non-singular.

Therefore, $A^{-1}$ exists.

Hence, the given system of equations is consistent.