On comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ , and $\frac{c_{1}}{c_{2}}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i) $5 x - 4 y + 8 = 0; 7 x + 6 y - 9 = 0$
(ii) $9 x + 3 y + 12 = 0; 18 x + 6 y + 24 = 0$
(iii) $6 x - 3 y + 10 = 0; 2 x - y + 9 = 0$

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Solution

If, for pair of equation, $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ ,
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , is true, then the lines are intersecting lines.
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ , is true, then the lines are coincident lines.
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ , is true, then the lines are parallel lines.
(i) $5 x - 4 y + 8 = 0; 7 x + 6 y - 9 = 0$
$\frac{a_{1}}{a_{2}} = \frac{5}{7}, \frac{b_{1}}{b_{2}} = \frac{- 4}{6}, \frac{c_{1}}{c_{2}} = \frac{8}{- 9}$
$∵ \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , the lines intersect at a point.
(ii) $9 x + 3 y + 12 = 0; 18 x + 6 y + 24 = 0$
$\frac{a_{1}}{a_{2}} = \frac{9}{18} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2}, \frac{c_{1}}{c_{2}} = \frac{12}{24} = \frac{1}{2}$
$∵ \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ , the lines are coincident.
(iii) $6 x - 3 y + 10 = 0; 2 x - y + 9 = 0$
$\frac{a_{1}}{a_{2}} = \frac{6}{2} = \frac{3}{1}, \frac{b_{1}}{b_{2}} = \frac{- 3}{- 1} = \frac{3}{1}, \frac{c_{1}}{c_{2}} = \frac{10}{9}$
$∵ \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ , the lines are parallel.

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\frac{a_{1}}{a_{2}}

\frac{b_{1}}{b_{2}}

and

\frac{c_{1}}{c_{2}}

, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident :
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5 x - 4 y + 8 = 0

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(ii)

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(i) $5 x - 4 y + 8 = 0$

$7 x + 6 y - 9 = 0$

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(iii) $6 x - 3 y + 10 = 0$

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\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}} a n d \frac{c_{1}}{c_{2}}

. find out whether the lines representing the following pairs of linear equations intersect at a pairs, are parallel or coincident:
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\begin{matrix} 5 x - 4 y + 8 = 0 7 x + 6 y - 9 = 0 \end{matrix}

(ii)

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\frac{a_{1}}{a_{2}}

\frac{b_{1}}{b_{2}}

and

\frac{c_{1}}{c_{2}}

, and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point and parallel or coincide:

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Q. Question 2 (ii)
On comparing the ratios

\frac{a_{1}}{a_{2}}

\frac{b_{1}}{b_{2}}

and

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, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
9x + 3y + 12 = 0
18x + 6y + 24 = 0

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On comparing the ratios a1a2,b1b2, and c1c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:(i) 5x−4y+8=0;7x+6y−9=0 (ii) 9x+3y+12=0;18x+6y+24=0(iii) 6x−3y+10=0;2x−y+9=0