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, are parallel or coincide:

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

Given:

Given pair of linear equations is:

$5x\ -\ 4y\ +\ 8\ =\ 0$

$7x\ +\ 6y\ -\ 9\ =\ 0$

To do:

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

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.

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Updated on: 10-Oct-2022

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