# On comparing the ratios a_{1}/a_{2}, b_{1}/b_{2} and c_{1}/c_{2}, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(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:**

For any pair of linear equation

a_{1} x + b_{1} y + c_{1} = 0

a_{2} x + b_{2} y + c_{2} = 0

a) a_{1}/a_{2} ≠ b_{1}/b_{2} (Intersecting Lines)

b) a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2} (Coincident Lines)

c) a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2} (Parallel Lines)

(i) 5x - 4 y + 8 = 0 and 7x + 6 y - 9 = 0

a_{1} = 5, b_{1} = - 4, c_{1} = 8

a_{2} = 7, b_{2} = 6, c_{2} = - 9

a_{1}/a_{2} = 5/7...(1)

b_{1}/b_{2}= -4/6 = -2/3...(2)

From (1) and (2)

a_{1}/a_{2} ≠ b_{1}/b_{2}

Therefore, they are intersecting lines at a point.

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

a_{1} = 9, b_{1} = 3, c_{1} = 12

a_{2} = 18, b_{2} = 6, c_{2} = 24

a_{1}/a_{2}= 9/18 = 1/2...(1)

b_{1}/b_{2}= 3/6 = 1/2...(2)

c_{1}/c_{2}= 12/24 = 1/2...(3)

From (1), (2) and (3)

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}= 1/2

Therefore, they are coincident lines.

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

a_{1} = 6, b_{1} = - 3, c_{1} = 10

a_{2} = 2, b_{2} = - 1, c_{2} = 9

a_{1}/a_{2 }= 6/2 = 3...(1)

b_{1}/b_{2 }= - 3/- 1 = 3...(2)

c_{1}/c_{2 }= 10/9...(3)

From (1), (2) and (3)

a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2}

Therefore, they are parallel lines.

**Video Solution:**

## On comparing the ratios a_{1}/a_{2}, b_{1}/b_{2} and c_{1}/c_{2}, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (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

### NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.2 Question 2:

On comparing the ratios a_{1}/a_{2}, b_{1}/b_{2} and c_{1}/c_{2}, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (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

On comparing the ratios of the coefficients of the given two equations, we can say that the lines are parallel to each other. (i) They are intersecting lines at a point. (ii) They are coincident lines. (iii) They are parallel lines