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:

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

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Video Solution:

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

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.

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☛ Related Questions:

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