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) $9 x + 3 y + 12 = 0$
$18 x + 6 y + 24 = 0$

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Solution

(i) $5 x - 4 y + 8 = 0$
$7 x + 6 y - 9 = 0$
Now, $\frac{a_{1}}{a_{2}} = \frac{5}{7}$ and $\frac{b_{1}}{b_{2}} = \frac{- 4}{6} = - \frac{2}{3}$
Since, $\frac{a_{1}}{a_{2}} \neq$ $\frac{b_{1}}{b_{2}}$ , above equations will have unique solution.
$∴$ the above lines will intersect at a point.

(ii) $9 x + 3 y + 12 = 0$
$18 x + 6 y + 24 = 0$
Now, $\frac{a_{1}}{a_{2}} = \frac{9}{18} = \frac{1}{2}$ , $\frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2}$ and $\frac{c_{1}}{c_{2}} = \frac{12}{24} = \frac{1}{2}$
Since, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ , above lines will be co-incident.

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Similar questions

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

Q. On comparing the ratios

\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:

9 x + 3 y + 12 = 0

18 x + 6 y + 24 = 0

Q. Question 2 (ii)
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.
9x + 3y + 12 = 0
18x + 6y + 24 = 0

Question 2 (i)
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.
5x - 4y + 8 = 0
7x + 6y - 9 = 0

On comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$ , and without drawing them, find out whether the lines representing the following pairs of linear equation intersect at a point, are parallel or coincide:

(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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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

(i) 5x−4y+8=0 7x+6y−9=0Now, a1a2=57 and b1b2=−46=−23Since, a1a2≠ b1b2, above equations will have unique solution.∴ the above lines will intersect at a point.(ii) 9x+3y+12=0 18x+6y+24=0Now, a1a2=918=12, b1b2=36=12 and c1c2=1224=12Since, a1a2=b1b2=c1c2, above lines will be co-incident.