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Solve the following pair of linear equations by the elimination method and the substitution method : (i) x + y = 5 and 2x – 3y = 4 (ii) 3x + 4y = 10 and 2x – 2y = 2 (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7 (iv)
(i) elimination method:
multiplying these 2 equations we get
subtracting (ii) from (iii) we get,
Substitution method,
from (i), we get,
putting this in (ii), we get,
(ii) Elimination method:
3x + 4y = 10 ...(1)
2x - 2y = 2 ...(2)
Multiplying equation (2) by 2, we obtain:
4x - 4y = 4 ...(3)
Adding equation (1) and (3), we obtain
7x = 14
x = 2
Substituting the value of x in equation (1), we obtain:
6 + 4y = 10
4y = 4
y = 1
Hence, x = 2, y = 1
Substitution method:
From equation (2), we obtain:
x = 1 + y ...(4)
Putting this value in equation (1), we obtain:
3(1 + y) + 4y = 10.
7y = 7
y = 1
Substituting the value of y in equation (4), we obtain:
(iii) elimination method
Multiplying the two equations, we get,
subtracting (iii) from (ii) we get, ,
Substituting the value of y in (i)
Substitution method:
From (i), we get,
placing in (ii),
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes if we only add 1 to the denominator. What is the fraction?
(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
(iv) Meena went to a bank to withdraw Rs. 2000. She asked the cashier to give her Rs. 50 and Rs. 100 notes only. Meena got 25 notes in all. Find how many notes of Rs. 50 and Rs. 100 she received.
(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs. 27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day
(i) Let the fraction be 
According to the question,
Subtracting equation (1) from equation (2), we get: x = 3
Substituting this value of x in equation (1), we get : , y=5
therefore the fraction is
(ii) Let present age of Nuri and Sonu be x and y respectively.
According to the question, .................(i)
Subtracting equation (1) from equation (2), we get:
y = 20
Substituting the value of y in equation (1), we get: , x=50
Thus, the age of Nuri and Sonu are 50 years and 20 years respectively.
(iii) Let the units digit and tens digit of the number be x and y respectively.
Number = 10y + x
Number after reversing the digits = 10x + y
According to the question,
x + y = 9 ... (1)
9(10y + x) = 2(10x + y)
88y - 11x = 0
- x + 8y =0 ... (2)
Adding equations (1) and (2), we obtain:
9y = 9
y = 1
Substituting the value of y in equation (1), we obtain:
x = 8
Thus, the number is 10y + x = 10 x 1 + 8 = 18
(iv) Let the number of Rs 50 notes and Rs 100 notes be x and y respectively.
According to the question,
Multiplying equation (1) by 50, we obtain:
50x + 50 y = 1250
Subtracting equation (3) from equation (2), we obtain:
50y = 750
y = 15
Substituting the value of y in equation (1), we obtain:
x = 10
Hence, Meena received 10 notes of Rs 50 and 15 notes of Rs 100.
(v) Let the fixed charge for first three days and each day charge thereafter be Rs x and Rs y respectively.
According to the question,
Subtracting equation (2) from equation (1), we obtain:
2y = 6
y = 3
Substituting the value of y in equation (1), we obtain:
x + 12 = 27
x = 15
Hence, the fixed charge is Rs 15 and the charge per day is Rs 3.
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