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Standard XII

Mathematics

Conditions for a system of linear equations to have no solutions

On comparing ...

Question

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:
$6 x - 3 y + 10 = 0$ $2 x - y + 9 = 0$

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Solution

$6 x - 3 y + 10 = 0 ⟶ (i) 2 x - y + 9 = 0 ⟶ (i i) 6 x - 3 y + 10 = 0$
Comparing with $a_{1} x + b_{1} y + c_{1} = 0$
We get ,
$a_{2} = 2, b_{2} = - 1, c_{2} = 9$
Now ,
$\frac{a_{1}}{a_{2}} = \frac{6}{2} = 3 \frac{b_{1}}{b_{2}} = \frac{- 3}{- 1} = 3 \frac{c_{1}}{c_{2}} = \frac{10}{9}$
$∴ \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
We have no solution.
Therefore , the lines that represent the linear equations are parellel.

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Similar questions

Q. Question 2 (iii)
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.
6x – 3y + 10 = 0
2x – 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, or are parallel or coincident.

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:

(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

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

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On comparing the ratios a1a2, b1b2 and c1c2, and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point and parallel or coincide:6x−3y+10=0 2x−y+9=0

6x−3y+10=0⟶(i)2x−y+9=0⟶(ii)6x−3y+10=0Comparing with a1x+b1y+c1=0We get , a2=2,b2=−1,c2=9Now , a1a2=62=3b1b2=−3−1=3c1c2=109∴a1a2=b1b2≠c1c2We have no solution.Therefore , the lines that represent the linear equations are parellel.

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:
$6 x - 3 y + 10 = 0$ $2 x - y + 9 = 0$