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# On comparing the ratios 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 - 6y = 9

(iii) 3/2x + 5/3y = 7; 9x -10y = 14

(iv) 5x - 3y = 11; -10x + 6y = -22

(v) 4/3x + 2y = 8; 2x + 3y = 12

**Solution:**

For any pair of linear equation

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Consistent means pair of linear equations will have an unique solution or infinitely many solutions.

a₁/a₂ ≠ b₁/b₂_{ }(Intersecting lines / unique Solution)

a₁/a₂ = b₁/b₂ = c₁/c₂_{ }(Coincident Lines / Infinitely many Solutions)

a₁/a₂ = b₁/b₂ ≠ c₁/c₂_{ }(Parallel lines / No Solution)

(i) 3x + 2y = 5; 2x - 3y = 7

a₁/a₂_{ }= 3/2

b₁/b₂ = 2/(-3)

c₁/c₂ = - 5/(-7) = 5/7

From the above,

a₁/a₂ ≠ b₁/b₂

Therefore, lines are intersecting and have a unique solution,

Hence, the pair of equations is consistent.

(ii) 2x - 3y = 8; 4x - 6y = 9

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = -3/(-6) = 1/2

c₁/c₂= -8/(-9) = 8/9

From the above,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, these lines are parallel and have no solution,

Hence, the pair of equations is inconsistent.

(iii) 3/2x + 5/3y = 7; 9x -10y = 14

a₁/a₂ = (3/2)/9 = (3/2) × (1/9) = 1/6

b₁/b₂ = (5/3)/(-10) = (5/3) × 1/(-10) = 1/(-6) = -1/6

From the above,

a₁/a₂ ≠ b₁/b₂

Therefore, lines are intersecting and have a unique solution.

Hence, they are consistent.

(iv) 5x - 3y = 11; -10x + 6y = -22

a₁/a₂ = 5/(-10) = -1/2

b₁/b₂ = -3/6 = -1/2

c₁/c₂ = -11/22 = -1/2

From the above,

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

(v) 4/3x + 2y = 8; 2x + 3y = 12

a₁/a₂_{ }= (4/3)/2 = (4/3) × (1/2) = 2/3

b₁/b₂_{ }= 2/3

c₁/c₂= -8/(-12) = 2/3

From the above,

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

**☛ 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 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

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

**Summary:**

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, we have seen whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2 y = 5; 2x - 3y = 7 consistent. (ii) 2x - 3y = 8; 4x - 6 y = 9 inconsistent. (iii) 3/2x + 5/3y = 7; 9x -10y = 14 consistent. (iv) 5x - 3y = 11; -10x + 6 y = -22 consistent. (v) 4/3x + 2 y = 8; 2x + 3y = 12 consistent.

**☛ Related Questions:**

- 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
- Draw the graphs of the equations-x - y + 1 = 0 and 3x + 2y - 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis and shade the triangular region

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