# 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.