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Last updated at March 11, 2020 by Teachoo

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Ex 3.2, 3 On comparing the ratios ๐1/๐2 , ๐1/๐2 & ๐1/๐2 , find out whether the following pair of linear equations are consistent, or inconsistent. 3x+ 2y= 5 ; 2xโ 3y= 7 3x + 2y โ 5 = 0 2x โ 3y โ 7 = 0 3x + 2y โ 5 = 0 Comparing with a1x + b1y + c1 = 0 โด a1 = 3 , b1 = 2 , c1 = โ5 2x โ 3y โ 7 = 0 Comparing with a2x + b2y + c2 = 0 โด a2 = 2 , b2 = โ3 , c2 = โ7 a1 = 3 , b1 = 2 , c1 = โ5 & a2 = 2 , b2 = โ3 , c2 = โ7 Since ๐1/๐2 โ ๐1/๐2 We have a unique solution Therefore, our system is consistent. ๐๐/๐๐ ๐๐/๐๐ ๐๐/๐๐ ๐1/๐2 = 3/2 ๐1/๐2 = 2/(โ3) ๐1/๐2 = -2/3 ๐1/๐2 = (โ5)/(โ7) ๐1/๐2 = 5/7 Ex 3.2, 3 On comparing the ratios ๐1/๐2 , ๐1/๐2 & ๐1/๐2 , find out whether the following pair of linear equations are consistent, or inconsistent. (ii) 2x -3y = 8 ; 4x - 6y = 9 2x โ 3y โ 8 = 0 4x โ 6y โ 9 = 0 2x โ 3y โ 8 = 0 Comparing with a1x + b1y + c1 = 0 โด a1 = 2 , b1 = โ 3 , c1 = โ 8 4x โ 6y โ 9 = 0 Comparing with a2x + b2y + c2 = 0 โด a2 = 4 , b2 = โ 6 , c2 = โ 9 โด a1 = 2, b1 = โ 3 , c1 = โ 8 a2 = 4, b2 = โ6, c2 = โ 9 Since ๐1/๐2 = ๐1/๐2 โ ๐1/๐2 We have no solution. Therefore, our system is inconsistent. ๐1/๐2 = 2/4 ๐1/๐2 = 1/2 ๐1/๐2 = (โ3)/(โ6) ๐1/๐2 = 1/2 ๐1/๐2 = (โ8)/(โ9) ๐1/๐2 = 8/9 ๐1/๐2 = (โ8)/(โ9) ๐1/๐2 = 8/9 Ex 3.2, 3 On comparing the ratios ๐1/๐2 , ๐1/๐2 & ๐1/๐2 , find out whether the following pair of linear equations are consistent, or inconsistent. (iii) 3/2x + 5/3y = 7 ; 9x - 10y = 14 3/2 ๐ฅ + 5/3 ๐ฆ โ 7 =0 9x โ 10y โ 14 = 0 ๐/๐ ๐ + ๐/๐ ๐ โ ๐ =๐ Comparing with a1x + b1y + c1 = 0 โด a1 = 3/2 , b1 = 5/3 , c1 = โ 7 9x โ 10y โ 14 = 0 Comparing with a2x + b2y + c2 = 0 โด a2 = 9 , b2 = โ 10, c2 = โ 14 ๐1/๐2 = (3/2)/9 ๐1/๐2 = 3/(2 ร 9) ๐1/๐2 = 1/(2 ๐ร3) ๐1/๐2 = 1/6 ๐1/๐2 = (5/3)/(โ10) ๐1/๐2 = 5/(โ3 ร10) ๐1/๐2 = 1/(โ3 ร 2) ๐1/๐2 = (โ1)/6 ๐1/๐2 = (โ7)/(โ14) ๐1/๐2 = 1/2 Since ๐1/๐2 โ ๐1/๐2 We have a unique solution Therefore, our system is consistent. Ex 3.2,3 On comparing the ratios ๐1/๐2 , ๐1/๐2 & ๐1/๐2 , find out whether the following pair of linear equations are consistent, or inconsistent. (iv) 5x โ 3y = 11 , -10x + 6y = -22 5x โ 3y โ 11 = 0 โ10x + 6y + 22 = 0 5x โ 3y โ 11 = 0 Comparing with a1x + b1y + c1 = 0 โด a1 = 5 , b1 = โ3 , c1 = โ 11 โ10x + 6y + 22 = 0 Comparing with a2x + b2y + c2 = 0 โด a2 = โ 10 , b2 = 6, c2 = 22 โด a1 = 5 , b1 = โ3 , c1 = โ 11 & a2 = โ 10 , b2 = 6, c2 = 22 Since ๐1/๐2 = ๐1/๐2 = ๐1/๐2 We have infinitely many solutions Therefore, our system is consistent. ๐๐/๐๐ ๐1/๐2 = 5/(โ10) ๐1/๐2 = โ1/2 ๐1/๐2 = (โ3)/6 ๐1/๐2 = โ1/2 ๐1/๐2 = (โ11)/22 ๐1/๐2 = โ 1/2 โด a1 = 5 , b1 = โ3 , c1 = โ 11 & a2 = โ 10 , b2 = 6, c2 = 22 Since ๐1/๐2 = ๐1/๐2 = ๐1/๐2 We have infinitely many solutions Therefore, our system is consistent. Ex 3.2, 3 On comparing the ratios ๐1/๐2 , ๐1/๐2 & ๐1/๐2 , find out whether the following pair of linear equations are consistent, or inconsistent. (v) 4/3 x + 2y = 8 ; 2x + 3y = 12 4/3 x + 2y โ 8 = 0 2x + 3y โ 12 = 0 ๐/๐ x + 2y โ 8 = 0 Comparing with a1x + b1y + c1 = 0 โด a1 = 4/3 , b1 = 2 , c1 = โ 8 2x + 3y โ 12 = 0 Comparing with a2x + b2y + c2 = 0 โด a2 = 2, b2 = 3 c2 = โ12 ๐1/๐2 = (4/3)/2 ๐1/๐2 = 4/(3 ร 2) ๐1/๐2 = 2/(3 ร 1) ๐1/๐2 = 2/3 ๐1/๐2 = 2/3 ๐1/๐2 = (โ8)/(โ12) ๐1/๐2 = 2/3 Since ๐1/๐2 = ๐1/๐2 = ๐1/๐2 We have infinite solutions. Therefore, our system is consistent.

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.