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

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Solution

The given equations are:

5x - 4y + 8 = 0
7x + 6y - 9 = 0

Comparing these equations with $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ , we get:
$a_{1} = 5, b_{1} = - 4, a n d c_{1} = 8$
$a_{2} = 7, b_{2} = 6 a n d c_{2} = - 9$

$\frac{a_{1}}{a_{2}} = \frac{5}{7},$

$\frac{b_{1}}{b_{2}} = \frac{- 4}{6}$ and

$\frac{c_{1}}{c_{2}} = \frac{8}{- 9}$

Hence, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

Therefore, the lines representing the given pair of linear equations intersect at a point.

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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

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

Q. 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

Q. 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.

6 x - 3 y + 10 = 0; 2 x - y + 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. 5x - 4y + 8 = 0 7x + 6y - 9 = 0

On comparing the ratios $\frac{a_{1}}{a_{2}}$ , $\frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$ , 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