Check whether the following pair of linear equation are consistent or inconsistent
$3 x + 2 y = 5$ , $2 x - 3 y = 7$ .

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Solution

Let $a_{1} x + b_{1} y + c_{1} = 3 x + 2 y - 5$ and $a_{2} x + b_{2} y + c_{2} = 2 x - 3 y - 7$

So, here $a_{1} = 3; b_{1} = 2; c_{1} = - 5; a_{2} = 2; b_{2} = - 3; c_{2} = - 7$

$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{3}{2}$

$\Rightarrow \frac{b_{1}}{b_{2}} = \frac{- 2}{3}$

$\Rightarrow \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

Hence the system of equations has unique solution.

Thus the pair of equations are consistent

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Check whether the following pair of linear equation are consistent or inconsistent 3x+2y=5, 2x−3y=7.

Let a1x+b1y+c1=3x+2y−5 and a2x+b2y+c2=2x−3y−7So, here a1=3; b1=2; c1=−5; a2=2; b2=−3; c2=−7⇒a1a2=32⇒b1b2=−23⇒a1a2≠b1b2Hence the system of equations has unique solution.Thus the pair of equations are consistent

Check whether the following pair of linear equation are consistent or inconsistent
$3 x + 2 y = 5$ , $2 x - 3 y = 7$ .