Ex 3.3, 3 (v) - Chapter 3 Class 10 Pair of Linear Equations in Two Variables

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Ex 3.3 ,3 Form the pair of linear equations for the following problems and find their solution by substitution method. (v) A fraction becomes 9/11 , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6 . Find the fraction. Let numerator be x and denominator be y So, fraction is 𝑥/𝑦 Given that if 2 is added to both the numerator and the denominator, fraction becomes 9/11. (𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 + 2)/(𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 + 2)=9/11 (𝑥 + 2)/(𝑦 + 2)=9/11 11(x + 2) = 9(y + 2) 11x + 22 = 9y + 18 11x – 9y + 22 – 18 = 0 11x – 9y + 4 = 0 Also, given that if 3 is added to both the numerator and the denominator, fraction becomes 5/6. (𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 + 3)/(𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 + 3)=5/6 (𝑥 + 3)/(𝑦 + 3)=5/6 6(x + 3) = 5(y + 3) 6(x + 3) = 5(y + 3) 6x + 18 = 5y + 15 6x – 5y + 18 – 15 = 0 6x – 5y + 3 = 0 Hence, our equations are 11x – 9y + 4 = 0 …(1) 6x – 5y + 3 = 0 …(2) From (1) 11x – 9y + 4 = 0 11x = 9y – 4 x = ((9𝑦 − 4)/11) Putting x in (2) 6x – 5y + 3 = 0 6 ((9𝑦 − 4)/11) – 5y + 3 = 0 Multiply both sides by 11 11 ×(6 (9𝑦−4))/11−5𝑦×11+3×11=0×11 6(9y – 4 ) – 55y + 33 = 0 6(9y) – 6(4) – 55y + 33 = 0 54y – 24 – 55y + 33 = 0 – y + 9 = 0 – y = –9 y = 9 Putting y = 9 in (1) 11x – 9y + 4 = 0 11x – 9(9) + 4 = 0 11x – 81 + 4 = 0 11x – 77 = 0 11x = 77 x = 77/11 x = 7 Therefore x = 7, y = 9 So, Numerator = x = 7 Denominator = y = 9 Hence, original fraction = 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟/𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 "= " 𝑥/𝑦=7/9