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AnilSinghBora
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On comparing the ratio, (a1/a2) , (b1/b2) , (c1/c2) find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7 (ii) 2x – 3y = 8 ; 4x – 6y = 9 (iii)(3/2)x+(5/3)y = 7; 9x – 10y = 14 (iv) 5x – 3y = 11 ; – 10x + 6y = –22 (v)(4/3)x+2y = 8 ; 2x + 3y = 12 Q.3

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The question of class 10th of exercise 3.2 of math subject. I want to know the answer of this question by easiest way. On comparing the ratio, (a1/a2) , (b1/b2) , (c1/c2) find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7 (ii) 2x – 3y = 8 ; 4x – 6y = 9 (iii)(3/2)x+(5/3)y = 7; 9x – 10y = 14 (iv) 5x – 3y = 11 ; – 10x + 6y = –22 (v)(4/3)x+2y = 8 ; 2x + 3y = 12

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  1. (i) Given : 3x + 2y = 5 or 3x + 2y -5 = 0

    and 2x – 3y = 7 or 2x – 3y -7 = 0

    Comparing these equations with a1x+b1y+c1 = 0

    And a2x+b2y+c2 = 0

    We get,

    a1 = 3, b1 = 2, c1 = -5

    a2 = 2, b2 = -3, c2 = -7

    (a1/a2) = 3/2

    (b1/b2) = 2/-3

    (c1/c2) = -5/-7 = 5/7

    Since, (a1/a2) ≠ (b1/b2)

    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

    Therefore,

    a1 = 2, b1 = -3, c1 = -8

    a2 = 4, b2 = -6, c2 = -9

    (a1/a2) = 2/4 = 1/2

    (b1/b2) = -3/-6 = 1/2

    (c1/c2) = -8/-9 = 8/9

    Since , (a1/a2) = (b1/b2) ≠ (c1/c2)

    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

    Therefore,

    a1 = 3/2, b1 = 5/3, c1 = -7

    a2 = 9, b2 = -10, c2 = -14

    (a1/a2) = 3/(2×9) = 1/6

    (b1/b2) = 5/(3× -10)= -1/6

    (c1/c2) = -7/-14 = 1/2

    Since, (a1/a2) ≠ (b1/b2)

    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

    Therefore,

    a1 = 5, b1 = -3, c1 = -11

    a2 = -10, b2 = 6, c2 = 22

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

    (b1/b2) = -3/6 = -1/2

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

    Since (a1/a2) = (b1/b2) = (c1/c2)

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

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

    a1 = 4/3 , b1= 2 , c1 = -8

    a2 = 2, b2 = 3 , c2 = -12

    (a1/a2) = 4/(3×2)= 4/6 = 2/3

    (b1/b2) = 2/3

    (c1/c2) = -8/-12 = 2/3

    Since (a1/a2) = (b1/b2) = (c1/c2)

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

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