7. 2x + 5y = 4 write another linear equation, so that lines represented by the pair are coincident.

Solution:

For any pair of linear equation,

a₁ x + b₁ y + c₁ = 0

a₂ x + b₂ y + c₂ = 0

a) If a₁/a₂ ≠ b₁/b₂ (Intersecting Lines)

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

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

(i) Intersecting lines

Condition: a₁/a₂ ≠ b₁/b₂

2x + 3y - 8 = 0

a₁ = 2

b₁ = 3

So, considering a₂ = 3 and b₂ = 2 will satisfy the condition for intersecting lines. c₂ can be any value.

a₁/a₂ = 2/3

b₁/b₂ = 3/2

2/3 ≠ 3/2

Therefore, another linear equation is 3x + 2y - 6 = 0

(ii) Parallel lines

Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

2x + 3y - 8 = 0

a₁ = 2

b₁ = 3

c₁ = - 8

So, considering a₂ = 4, b₂ = 6, c₂ = 9 will satisfy the condition for parallel lines.

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = - 8/9

Thus, a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, another linear equation is 4x + 6y + 9 = 0

(iii) Coincident lines

Condition: a₁/a₂ = b₁/b₂ = c₁/c₂

2x + 3y - 8 = 0

Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

2x + 3y - 8 = 0

We know that, a₁= 2, b₁= 3, c₁= - 8

So, considering a₂ = 4, b₂ = 6, c₂ = - 16 will satisfy the condition for coincident lines.

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = - 8/(-16) = 1/2

Thus, a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, linear equation is 4x + 6y -16 = 0

☛ Check: Class 10 Maths NCERT Solutions Chapter 3

Video Solution:

NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.2 Question 6

Summary:

Given the linear equation 2x + 3y - 8 = 0, another linear equation in two variables such that the geometrical representation of the pair so formed is intersecting lines is 3x + 2y - 6 = 0, for parallel lines is 4x + 6y + 9 = 0 and for the coincident lines is 4x + 6y - 16 = 0.

☛ Related Questions:

• On comparing the ratios a1/a2 = b1/b2 = 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 
• 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.

• Pair of Linear Equations in Two Variables

Ans) 1) intersecting = 4x-3y-4=0         ​ 2) coincident = 4x-10y-8=0       ​   3) parallel = 2x-5y-2=0 ​

​Hope it helps.