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

To do:

We have to find whether the lines representing the given pairs of linear equations intersect at a point, are parallel or coincident.

Solution:

Comparing the given pair of linear equations with the standard form of linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, we get,

$a_1=5, b_1=-4$ and $c_1=8$

$a_2=7, b_2=6$ and $c_2=-9$

Here,

$\frac{a_1}{a_2}=\frac{5}{7}$

$\frac{b_1}{b_2}=\frac{-4}{6}=\frac{-2}{3}$

$\frac{c_1}{c_2}=\frac{8}{-9}$

$\frac{a_1}{a_2} ≠ \frac{b_1}{b_2}$

Therefore, the two lines intersect each other at a point.

(ii) Comparing the given pair of linear equations with the standard form of linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, we get,

$a_1=9, b_1=3$ and $c_1=12$

$a_2=18, b_2=6$ and $c_2=24$

Here,

$\frac{a_1}{a_2}=\frac{9}{18}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{12}{24}=\frac{1}{2}$

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Therefore, the two lines coincide with each other. 

(iii) Comparing the given pair of linear equations with the standard form of linear equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, we get,

$a_1=6, b_1=-3$ and $c_1=10$

$a_2=2, b_2=-1$ and $c_2=9$

Here,

$\frac{a_1}{a_2}=\frac{6}{2}=3$

$\frac{b_1}{b_2}=\frac{-3}{-1}=3$

$\frac{c_1}{c_2}=\frac{10}{9}$

$\frac{a_1}{a_2} = \frac{b_1}{b_2} ≠ \frac{c_1}{c_2}$

Therefore, the two lines are parallel to each other. 

Tutorialspoint

Updated on: 10-Oct-2022

39 Views

Kickstart Your Career

Get certified by completing the course

Get Started