Write whether the following pair of linear equation is consistent or inconsistent 3x 6x 4y 11

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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/uniqueSolution)

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

c) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Parallel Lines/No solution)

(i) x + y = 5, 2x + 2y = 10

a₁/a₂= 1/2

b₁/b₂= 1/2

c₁/c₂= -5/(-10) = 1/2

From the above,

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, lines are coincident and have infinitely many solutions. Hence, they are consistent.

x + y - 5 = 0

y = - x + 5

y = 5 - x

2x + 2y - 10 = 0

2y = 10 - 2x

y = 5 - x

All the points on coincident line are solutions for the given pair of equations.

(ii) x - y = 8, 3x - 3y =16

a₁/a₂ = 1/3

b₁/b₂ = -1/(-3) = 1/3

c₁/c₂ = - 8/(-16) = 1/2

From the above,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, lines are parallel and have no solution.

Hence, the pair of equations are inconsistent.

(iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = 1/(-2) = -1/2

c₁/c₂ = -6/(-4) = 3/2

From the above,

a₁/a₂ ≠ b₁/b₂

Therefore, lines are intersecting and have a unique solution.

Hence, they are consistent.

2x + y - 6 = 0

y = 6 - 2x

4x - 2y - 4 = 0

2y = 4x - 4

y = 2x - 2

x = 2 and y = 2 are solutions for the given pair of equations.

(iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = -2/(-4) = 1/2

c₁/c₂ = -2/(-5) = 2/5

From the above,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, lines are parallel and have no solution.

Hence, the pair of equations are inconsistent.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 3

Video Solution:

Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10 (ii) x - y = 8, 3x - 3y =16 (iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0 (iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0

NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.2 Question 4

Summary:

On comparing the ratios of the coefficients of the following pairs of linear equations, we see that (i) x + y = 5, 2x + 2y = 10 have infinitely many solutions. Hence, they are consistent. (ii) x - y = 8, 3x - 3y =16 are parallel and have no solution.Hence, the pair of equations are inconsistent. (iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0 are intersecting and have a unique solution. Hence, they are consistent. (iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0 are parallel and have no solution. Hence, the pair of equations are inconsistent.

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Find whether the following pair of linear equations is consistent or inconsistent:3 x+2 y=86 x 4 y=9

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