(i) 3x + 2y = 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:**

**(i) 3x + 2y = 5 or 3x + 2y -5 = 0**

**and 2x – 3y = 7 or 2x – 3y -7 = 0**

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

a_{1} = 3, b_{1} = 2, c_{1} = -5

a_{2} = 2, b_{2} = -3, c_{2} = -7

(a_{1}/a_{2}) = 3/2

(b_{1}/b_{2}) = 2/-3

(c_{1}/c_{2}) = -5/-7 = 5/7

Since, (a_{1}/a_{2}) ≠ (b_{1}/b_{2})

So, the given equations intersect each other at one point and they have only one possible solution. The equations are consistent.

**(ii) Given 2x – 3y = 8 and 4x – 6y = 9 **

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

a_{1} = 2, b_{1} = -3, c_{1} = -8

a_{2} = 4, b_{2} = -6, c_{2} = -9

(a_{1}/a_{2}) = 2/4 = 1/2

(b_{1}/b_{2}) = -3/-6 = 1/2

(c_{1}/c_{2}) = -8/-9 = 8/9

Since , (a_{1}/a_{2}) = (b_{1}/b_{2}) ≠ (c_{1}/c_{2})

So, the equations are parallel to each other and they have no possible solution. Hence, the equations are inconsistent.

**(iii) Given (3/2)x + (5/3)y = 7 and 9x – 10y = 14 **

Comparing these equations with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

a_{1} = 3/2, b_{1} = 5/3, c_{1} = -7

a_{2} = 9, b_{2} = -10, c_{2} = -14

(a_{1}/a_{2}) = 3/(2×9) = 1/6

(b_{1}/b_{2}) = 5/(3× -10)= -1/6

(c_{1}/c_{2}) = -7/-14 = 1/2

Since, (a_{1}/a_{2}) ≠ (b_{1}/b_{2})

So, the equations are intersecting each other at one point and they have only one possible solution. Hence, the equations are consistent.

**(iv) Given, 5x – 3y = 11 and – 10x + 6y = –22 **

_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

a_{1} = 5, b_{1} = -3, c_{1} = -11

a_{2} = -10, b_{2} = 6, c_{2} = 22

(a_{1}/a_{2}) = 5/(-10) = -5/10 = -1/2

(b_{1}/b_{2}) = -3/6 = -1/2

(c_{1}/c_{2}) = -11/22 = -1/2

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})

These linear equations are coincident lines and have an infinite number of possible solutions. Hence, the equations are consistent.

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

_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we get,

a_{1} = 4/3 , b_{1}= 2 , c_{1} = -8

a_{2} = 2, b_{2} = 3, c_{2 }= -12

(a_{1}/a_{2}) = 4/(3×2)= 4/6 = 2/3

(b_{1}/b_{2}) = 2/3

(c_{1}/c_{2}) = -8/-12 = 2/3

Since (a_{1}/a_{2}) = (b_{1}/b_{2}) = (c_{1}/c_{2})

These linear equations are coincident lines and have an infinite number of possible solutions. Hence, the equations are consistent.