Question

On comparing the ratios $\dfrac{{{a_1}}}{{{a_2}}},\dfrac{{{b_1}}}{{{b_2}}}$ and $\dfrac{{{c_1}}}{{{c_2}}}$ and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.
6x - 3y + 10=0 and 2x -y + 9=0

Answer 1

Hint: The following relations are to be used and checked:
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$…… (Coincident lines)
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$……. (Parallel lines)
$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$…………… (Intersecting lines)

Complete step-by-step answer:
Here in the question Equation 1 is (6x - 3y + 10=0)
Here, \[{a_1} = 6,{b_1} = - 3,{c_1} = 10\]
And now the given equation 2 is (2x -y + 9=0)
Here, \[{a_2} = 2,{b_2} = - 1,{c_2} = 9\]
Comparing the ratios:
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{6}{2} = 3,\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{ - 3}}{{ - 1}} = 3,\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{10}}{9}$
From here we can see that it satisfies the condition which is parallel lines I.e. $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
Hence the lines representing the following pairs of linear equations are parallel.

Note: Whenever we face such a type of question the key concept for solving the question is to remember the relations between the ratios of their coefficients to solve these types of questions. Also, keep in mind while finding the ratios the sign with the coefficient should also be considered.