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3. 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

3. 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

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 8740 Points
4 months ago
Solutions: (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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