The graph for the given pair of linear equations in two variables when a1/a2 = b1/b2 = c1/c2

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Ex 3.2, 2 On comparing the ratios 𝑎1/𝑎2 , 𝑏1/𝑏2 & 𝑐1/𝑐2 , find out whether the lines representing the following pair of linear equations intersect at a point, parallel or coincident 5x – 4y + 8 = 0 ; 7x + 6y – 9 = 0 5x – 4y + 8 = 0 7x + 6y – 9 = 0 5x – 4y + 8 = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = 5 , b1 = −4 , c1 = 8 7x + 6y – 9 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = 7 , b2 = 6 , c2 = −9 ∴ a1 = 5 , b1 = −4 , c1 = 8 & a2 = 7 , b2 = 6 , c2 = −9 𝒂𝟏/𝒂𝟐 𝑎1/𝑎2 = 5/7 𝒃𝟏/𝒃𝟐 𝑏1/𝑏2 = (−4)/6 𝑏1/𝑏2 = (−2)/3 𝒄𝟏/𝒄𝟐 𝑐1/𝑐2 = 8/(−9) 𝑐1/𝑐2 = (−8)/9 Since 𝑎1/𝑎2 ≠ 𝑏1/𝑏2 So, have unique solution Therefore, the lines that represent the linear equations intersect at a point

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