# On comparing the ratios find out whether the following pair of linear a_{1}/a_{2}, b_{1}/b_{2 and }c_{1}/c_{2}, find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2 y = 5; 2x - 3y = 7

(ii) 2x - 3y = 8; 4x - 6 y = 9

(iii) 3/2x + 5/3y = 7; 9x -10y = 14

(iv) 5x - 3y = 11; -10x + 6 y = -22

(v) 4/3x + 2 y = 8; 2x + 3y = 12

**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

Consistent means pair of linear equations have one solution or infinitely many solutions.

a_{1}/a_{2} = b_{1}/b_{2 }(Intersecting lines / one Solution)

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2 }(Coincident Lines / Infinitely many Solutions)

a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2 }(Parallel lines / No Solution)

(i) 3x + 2 y = 5; 2x - 3y = 7

a_{1}/a_{2 }= 3/2

b_{1}/b_{2}= 2/-3

c_{1}/c_{2}= -5/-7 = 5/7

From above

a_{1}/a_{2} ≠ b_{1}/b_{2}

Therefore, lines are intersecting and have one solution,

Hence, the pair of equations are consistent.

(ii) 2x - 3y = 8; 4x - 6 y = 9

a_{1}/a_{2}= 2/4 = 1/2

b_{1}/b_{2}= -3/-6 = 1/2

c_{1}/c_{2}= -8/-9 = 8/9

From above

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

Therefore, these lines are parallel and have no solution,

Hence, the pair of equations is inconsistent.

(iii) 3/2x + 5/3y = 7; 9x -10y = 14

a_{1}/a_{2}= (3/2)/9 = (3/2) × (1/9) = 1/6

b_{1}/b_{2}= (5/3)/-10 = (5/3) × (1/-10) = 1/-6

c_{1}/c_{2}= 7/14 = 1/2

From above

a_{1}/a_{2} ≠ b_{1}/b_{2}

Therefore, lines are intersecting and have one solution.

Hence, they are consistent.

(iv) 5x - 3y = 11; -10x + 6 y = -22

a_{1}/a_{2}= 5/-10 = -1/2

b_{1}/b_{2}= -3/6 = -1/2

c_{1}/c_{2}= -11/22 = -1/2

From above

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

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

(v) 4/3x + 2 y = 8; 2x + 3y = 12

a_{1}/a_{2 }= (4/3)/2 = (4/3) × (1/2) = 2/3

b_{1}/b_{2 }= 2/3

c_{1}/c_{2}= -8/-12 = 2/3

From above

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

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

**Video Solution:**

## On comparing the ratios find out whether the following pair of linear a1/a2,b1/b2 and c1/c2, find out whether the following pair of linear equations are consistent, or inconsistent (i) 3x + 2 y = 5; 2x - 3y = 7 (ii) 2x - 3y = 8; 4x - 6 y = 9 (iii) 3/2x + 5/3y = 7; 9x -10y = 14 (iv) 5x - 3y = 11; -10x + 6 y = -22 (v) 4/3x + 2 y = 8; 2x + 3y = 12

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

On comparing the ratios find out whether the following pair of linear a1/a2,b1/b2 and c1/c2, find out whether the following pair of linear equations are consistent, or inconsistent (i) 3x + 2 y = 5; 2x - 3y = 7 (ii) 2x - 3y = 8; 4x - 6 y = 9 (iii) 3/2x + 5/3y = 7; 9x -10y = 14 (iv) 5x - 3y = 11; -10x + 6 y = -22 (v) 4/3x + 2 y = 8; 2x + 3y = 12

On comparing the ratios of the coefficients, we can say that the equations have infinitely many solutions. (i) consistent. (ii) inconsistent. (iii) consistent. (iv) consistent. (v) consistent.