Ex 3.2, 2 - Chapter 3 Class 10 Pair of Linear Equations in Two Variables

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Ex 3.2, 2 On comparing the ratios 𝑎1/𝑎2 , 𝑏1/𝑏2 & 𝑐1/𝑐2 , find out whether the lines representing the following pair of linear equations intersect at a point, parallel or coincident 5x – 4y + 8 = 0 ; 7x + 6y – 9 = 0 5x – 4y + 8 = 0 7x + 6y – 9 = 0 5x – 4y + 8 = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = 5 , b1 = −4 , c1 = 8 7x + 6y – 9 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = 7 , b2 = 6 , c2 = −9 ∴ a1 = 5 , b1 = −4 , c1 = 8 & a2 = 7 , b2 = 6 , c2 = −9 𝒂𝟏/𝒂𝟐 𝑎1/𝑎2 = 5/7 𝒃𝟏/𝒃𝟐 𝑏1/𝑏2 = (−4)/6 𝑏1/𝑏2 = (−2)/3 𝒄𝟏/𝒄𝟐 𝑐1/𝑐2 = 8/(−9) 𝑐1/𝑐2 = (−8)/9 Since 𝑎1/𝑎2 ≠ 𝑏1/𝑏2 So, have unique solution Therefore, the lines that represent the linear equations intersect at a point Ex 3.2, 2 On comparing the ratios 𝑎1/𝑎2 , 𝑏1/𝑏2 & 𝑐1/𝑐2 , find out whether the lines representing the following pair of linear equations intersect at a point, parallel or coincident (ii) 9x + 3y + 12 = 0 ; 18x + 6y + 24= 0 9x + 3y + 12 = 0 18x + 6y + 24 = 0 9x + 3y + 12 = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = 9 , b1 = 3 , c1 = 12 18x + 6y + 24 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = 18 , b2 = 6 , c2 = 24 ∴ a1 = 9 , b1 = 3 , c1 = 12 & a2 = 18 , b2 = 6 , c2 = 24 𝒂𝟏/𝒂𝟐 𝑎1/𝑎2 = 9/18 𝑎1/𝑎2 = 1/2 𝒃𝟏/𝒃𝟐 𝑏1/𝑏2 = 3/6 𝑏1/𝑏2 = 1/2 𝒄𝟏/𝒄𝟐 𝑐1/𝑐2 = 12/24 𝑐1/𝑐2 = 1/2 Since 𝑎1/𝑎2 = 𝑏1/𝑏2 = 𝑐1/𝑐2 So, we have infinite solutions Therefore, the lines that represent the linear equations are coincident Ex 3.2, 2 On comparing the ratios 𝑎1/𝑎2 , 𝑏1/𝑏2 & 𝑐1/𝑐2 , find out whether the lines representing the following pair of linear equations intersect at a point, parallel or coincident (iii) 6x – 3y + 10 = 0 ; 2x – y + 9 = 0 6x – 3y + 10 = 0 2x – y + 9 = 0 6x – 3y + 10 = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = 6 , b1 = −3 , c1 = 10 2x – y + 9 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = 2 , b2 = −1 , c2 = 9 ∴ a1 = 6 , b1 = −3 , c1 = 10 & a2 = 2 , b2 = −1 , c2 = 9 𝒂𝟏/𝒂𝟐 𝑎1/𝑎2 = 6/2 𝑎1/𝑎2 = 3 𝒃𝟏/𝒃𝟐 𝑏1/𝑏2 = (−3)/(−1) 𝑏1/𝑏2 = 3 𝒄𝟏/𝒄𝟐 𝑐1/𝑐2 = 10/9 Since 𝑎1/𝑎2 = 𝑏1/𝑏2 ≠ 𝑐1/𝑐2 So, we have no solution Therefore, the lines represent the linear equations are parallel