For what value of p the pair of linear equation 2x 3y 7 and 4x 6y C 0 are coincident A 14 B 14 C 21 D 7 2?

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, 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

a₁ x + b₁ y + c₁ = 0

a₂ x + b₂ y + c₂ = 0

a) a₁/a₂ ≠ b₁/b₂ (Intersecting Lines)

b) a₁/a₂ = b₁/b₂ = c₁/c₂ (Coincident Lines)

c) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Parallel Lines)

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

a₁ = 5, b₁ = - 4, c₁ = 8

a₂ = 7, b₂ = 6, c₂ = - 9

a₁/a₂ = 5/7...(1)

b₁/b₂ = -4/6 = -2/3...(2)

From (1) and (2)

a₁/a₂ ≠ b₁/b₂

Therefore, they are intersecting lines at a point.

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

a₁ = 9, b₁ = 3, c₁ = 12

a₂ = 18, b₂ = 6, c₂ = 24

a₁/a₂ = 9/18 = 1/2...(1)

b₁/b₂ = 3/6 = 1/2...(2)

c₁/c₂ = 12/24 = 1/2...(3)

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

a₁/a₂ = b₁/b₂ = c₁/c₂= 1/2

Therefore, they are coincident lines.

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

a₁ = 6, b₁ = - 3, c₁ = 10

a₂ = 2, b₂ = - 1, c₂ = 9

a₁/a₂ = 6/2 = 3...(1)

b₁/b₂ = - 3/(- 1 ) = 3...(2)

c₁/c₂ = 10/9...(3)

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

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, they are parallel lines.

☛ Check: Class 10 Maths NCERT Solutions Chapter 3

Video Solution:

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, 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

Summary:

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, we have seen whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident to each other as follows:  (i) 5x – 4y + 8 = 0 7x + 6y – 9 = 0 they are intersecting lines at a point. (ii) 9x + 3y + 12 = 0 18x + 6y + 24 = 0 they are coincident lines. (iii) 6x – 3y + 10 = 0 2x – y + 9 = 0 they are parallel lines.

☛ Related Questions:

• 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
• Which of the following pairs of linear equations are consistent / inconsistent? If consistent, obtain the Solution graphically: (i) x + y = 5, 2x + 2 y = 10 (ii) x - y = 8, 3x - 3y =16 (iii) 2x + y - 6 = 0, 4x - 2 y - 4 = 0 (iv) 2x - 2 y - 2 = 0, 4x - 4 y - 5 = 0
• Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
• Given the linear equation 2x + 3y - 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines (ii) parallel lines (iii) coincident lines

Solution:

Given, the pair of linear equations is

λx + 3y = -7

2x + 6y = 14

We have to determine if λ is 1 the pair of equations has infinitely many solutions.

We know that,

For a pair of linear equations in two variables be a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0,

If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\), then

i) the pair of linear equation is dependent and consistent

ii) the graph will be a pair of coincident lines. Each point on the lines will be a solution and so the pair of equations will have infinitely many solutions.

Here, a1 = λ, b1 = 3, c1 = -7

a2 = 2, b2 = 6, c2 = 14

So, a1/a2 = λ/2

b1/b2 = 3/6 = 1/2

c1/c2 = -7/14 = -1/2

Case a) λ/2 = 1/2

λ = 2/2

λ = 1

Case b) λ/2 = -1/2

λ = -2/2

λ = -1

The value of λ is not constant.

Therefore, the pair of equations to have infinitely many solutions the value of λ should not be 1.

✦ Try This: For the pair of equations λx + 2y = 7; 2x + 4y = 14 to have infinitely many solutions, the value of λ should be 1. Is the statement true?

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3

NCERT Exemplar Class 10 Maths Exercise 3.2 Problem 4

For the pair of equations λx + 3y = -7; 2x + 6y = 14 to have infinitely many solutions, the value of λ should be 1. Is the statement true?

Summary:

For the pair of equations λx + 3y = -7; 2x + 6y = 14 to have infinitely many solutions, the value of λ should be 1. The statement is not true.

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