Prove that the pair of lines represented by equations $2 x - 3 y + 5 = 0$ and $4 x - 6 y + 7 = 0$ are inconsistent.

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Solution

REF.Image
$l_{1} \equiv 2 x - 3 y + 5 = 0$ Graphically
Comparing with $a_{1} x + b_{1} y + c_{1} = 0$ $y = \frac{2 x + 5}{3}$ ___________ (1)
$a_{1} = 2$ $b_{1} = - 3$ $c = 5$ $y = \frac{4 x + 7}{6}$ ____________ (2)
$l_{2} \equiv 4 x - 6 y + 7 = 0$
condition of inconsistent lines is
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$∴ \frac{2}{4} = \frac{- 3}{- 6} \neq \frac{5}{7}$ $∴ \frac{1}{2} = \frac{1}{2} \neq \frac{5}{7}$
Since, above both equations satisfy the relation, therefore given lines are inconsistent.

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Prove that the pair of lines represented by equations 2x−3y+5=0 and 4x−6y+7=0 are inconsistent.

Prove that the pair of lines represented by equations $2 x - 3 y + 5 = 0$ and $4 x - 6 y + 7 = 0$ are inconsistent.