On comparing the ratio $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}} a n d \frac{c_{1}}{c_{2}}$ . find out whether the lines representing the following pairs of linear equations intersect at a pairs, are parallel or coincident:
(i) $\begin{matrix} 5 x - 4 y + 8 = 0 7 x + 6 y - 9 = 0 \end{matrix}$
(ii) $\begin{matrix} 9 x + 3 y + 12 = 0 18 x + 6 y + 24 = 0 \end{matrix}$
(iii) $\begin{matrix} 6 x - 3 y + 10 = 0 2 x - y + 9 = 0 \end{matrix}$

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Solution

Part (1).

$5 x - 4 y + 8 = 0$ ……. (1)

$7 x + 6 y - 9 = 0$ ……. (2)

Comparing equation (1) and (2) to,

$a_{1} x + b_{1} y + c_{1} = 0$

$a_{2} x + b_{2} y + c_{2} = 0$

Then,

$a_{1} = 5, b_{1} = - 4, c_{1} = 8$

$a_{2} = 7, b_{2} = 6, c_{2} = - 9$

Now, we know that,

$\frac{a_{1}}{a_{2}} = \frac{5}{7}, \frac{b_{1}}{b_{2}} = \frac{- 4}{6} = \frac{- 2}{3}, \frac{c_{1}}{c_{2}} = \frac{8}{- 9}$

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

We have a unique solution.

Therefore, the linear equation intersect at a point.

Part (2).

$9 x + 3 y + 12 = 0$ ……. (1)

$18 x + 6 y + 24 = 0$ ……. (2)

Comparing equation (1) and (2) to,

$a_{1} x + b_{1} y + c_{1} = 0$

$a_{2} x + b_{2} y + c_{2} = 0$

Then,

$a_{1} = 9, b_{1} = 3, c_{1} = 12$

$a_{2} = 18, b_{2} = 6, c_{2} = 24$

Now, we know that,

$\frac{a_{1}}{a_{2}} = \frac{9}{18} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2}, \frac{c_{1}}{c_{2}} = \frac{12}{24} = \frac{1}{2}$

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$

We have a infinite solution.

Therefore, the linear equation are coincident.

Part (3).

$6 x - 3 y + 10 = 0$ ……. (1)

$2 x - y + 9 = 0$ ……. (2)

Comparing equation (1) and (2) to,

$a_{1} x + b_{1} y + c_{1} = 0$

$a_{2} x + b_{2} y + c_{2} = 0$

Then,

$a_{1} = 6, b_{1} = - 3, c_{1} = 10$

$a_{2} = 2, b_{2} = - 1, c_{2} = 9$

Now, we know that,

$\frac{a_{1}}{a_{2}} = \frac{6}{2} = \frac{3}{1}, \frac{b_{1}}{b_{2}} = \frac{- 3}{- 1} = \frac{3}{1}, \frac{c_{1}}{c_{2}} = \frac{10}{9}$
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

We have no solution.

Therefore, the linear equation are parallel.

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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}}

, 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

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: