Question

On comparing the ratios a1/a2, b1/b2 and c1/c2, find out whether the following pairs of linear equations are consistent or inconsistent.

1. 3x/2 + 5y/3 = 7; 9x - 10y = 14
2. 5x - 4y + 8 = 0; 7x + y - 9 = 0
3. 4x/3 + 2y = 8; 2x + 3y = 12
4. 9x + 3y + 12 = 0; 18x + 6y + 24 = 0.​

Answer 1

(i) 3x + 2y = 5 ; 2x – 3y = 7

a1/a2 = 3/2

b1/b2 = -2/3 and

c1/c2 = 5/7

Hence, a1/a2 ≠ b1/b2

These linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(ii) 2x – 3y = 8 ; 4x – 6y = 9

a1/a2 = 2/4 = 1/2

b1/b2 = -3/-6 = 1/2 and

c1/c2 = 8/9

Hence, a1/a2 = b1/b2 ≠ c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii) 3/2x + 5/3y = 7 ; 9x – 10y = 14

a1/a2 = 3/2/9 = 1/6

b1/b2 = 5/3/-10 = -1/6 and

c1/c2 = 7/14 = 1/2

Hence, a1/a2 ≠ b1/b2

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(iv) 5x – 3y = 11 ; – 10x + 6y = –22

a1/a2 = 5/-10 = -1/2

b1/b2 = -3/6 = -1/2 and

c1/c2 = 11/-22 = -1/2

Hence, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

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

a1/a2 = 4/3/2 = 2/3

b1/b2 = /3 and

c1/c2 = 8/12 = 2/3

Hence, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

Answer 2

Answer:

i) 3x + 2y = 5 ; 2x – 3y = 7

a1/a2 = 3/2

b1/b2 = -2/3 and

c1/c2 = 5/7

Hence, a1/a2 ≠ b1/b2

These linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(ii) 2x – 3y = 8 ; 4x – 6y = 9

a1/a2 = 2/4 = 1/2

b1/b2 = -3/-6 = 1/2 and

c1/c2 = 8/9

Hence, a1/a2 = b1/b2 ≠ c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii) 3/2x + 5/3y = 7 ; 9x – 10y = 14

a1/a2 = 3/2/9 = 1/6

b1/b2 = 5/3/-10 = -1/6 and

c1/c2 = 7/14 = 1/2

Hence, a1/a2 ≠ b1/b2

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(iv) 5x – 3y = 11 ; – 10x + 6y = –22

a1/a2 = 5/-10 = -1/2

b1/b2 = -3/6 = -1/2 and

c1/c2 = 11/-22 = -1/2

Hence, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

(v) 4/3x + 2y =8 ; 2x + 3y = 12

a1/a2 = 4/3/2 = 2/3

b1/b2 = /3 and

c1/c2 = 8/12 = 2/3

Hence, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.