On comparing the ratios \(a_1\over a_2\), \(b_1\over b_2\) and \(c_1\over c_2\), find out whether the lines representing the following pair of linear equations intersect at a point, are parallel of coincide.

Question : On comparing the ratios \(a_1\over a_2\), \(b_1\over b_2\) and \(c_1\over c_2\), find out whether the lines representing the following pair of linear equations intersect at a point, are parallel of coincide.

(i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0

(ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0

Solution :

(i)  Linear Equations given are

5x – 4y + 8 = 0

and  7x + 6y – 9 = 0

Here, \(a_1\over a_2\) \(\ne\) \(b_1\over b_2\)

i.e.  \(5\over 7\) \(\ne\) \(-4\over 6\)

So, given lines are intersecting lines.

(ii)  Linear Equations given are

9x + 3y + 12 = 0

and  18x + 6y + 24 = 0

Here, \(a_1\over a_2\) = \(b_1\over b_2\) = \(c_1\over c_2\)

i.e.  \(9\over 18\) = \(3\over 6\) = \(12\over 24\)

So, given lines are coincident lines.

(iii)  Linear Equations given are

6x – 3y + 10 = 0

and  2x – y + 9 = 0

Here, \(a_1\over a_2\) = \(b_1\over b_2\) \(\ne\) \(c_1\over c_2\)

i.e.  \(6\over 2\) = \(-3\over -1\) \(\ne\) \(10\over 9\)

So, given lines are parallel lines.