Determine graphically whether the following pair of linear equations $2 x - 3 y = 5$ , $3 x + 4 y = - 1$ has (a) a unique solution (b) infinitely many solutions or (c) no solution.

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Solution

$2 x - 3 y = 5 \to 2 x - 3 y - 5 = 0$
$3 x + 4 y = - 1 \to 3 x + 4 y + 1 = 0$
on comparing theses two equation with $a x + b y + c = 0$
$a_{1} = 2, b_{1} = - 3, c_{1} = - 5$
$a_{2} = 3, b_{2} = 4, c_{2} = 1$
$\frac{a_{1}}{a_{2}} = \frac{2}{3} \frac{b_{1}}{b_{2}} = \frac{- 3}{4} \frac{c_{1}}{c_{2}} = - 5$
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
Therefore given pair of lines has unique solution.

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Determine graphically whether the following pair of linear equations 2x−3y=5, 3x+4y=−1 has (a) a unique solution (b) infinitely many solutions or (c) no solution.

2x−3y=5→2x−3y−5=03x+4y=−1→3x+4y+1=0on comparing theses two equation with ax+by+c=0a1=2,b1=−3,c1=−5a2=3,b2=4,c2=1a1a2=23b1b2=−34c1c2=−5a1a2≠b1b2Therefore given pair of lines has unique solution.

Determine graphically whether the following pair of linear equations $2 x - 3 y = 5$ , $3 x + 4 y = - 1$ has (a) a unique solution (b) infinitely many solutions or (c) no solution.