**Pair of Linear Equations in Two Variables**

**Question 1.Solve the following pairs of linear equations by the elimination method and the substitution method:**

**Solution:**

**(i)** By Elimination Method:

Fquations are x + y = 5

and 2x – 3y = 4

Multiply equation (i) by 2 and subtract equation (ii) from it, we have**(ii)** By Elimination method:

Equations are 3x + 4y = 10

and 2x – 2y = 2

Multiplying equation (ii) by 2 and adding to equation (i), we

**By Elimination Method:**

(iii)

(iii)

**(iv)** By Elimination Method:**1st equation :**

**Question 2.Form the pair of linear equations for 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 ₹2000. She asked the cashier to give her ₹50 and ₹100 notes only. Meena got 25 notes in all. Find how many notes of ₹50 and ₹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 ₹ 27 for a book kept for seven days, while Susy paid ₹21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

**Solution:**

**(i)** Let numerator be x and denominator be y.

Fraction = x/y**A.T.Q.(ii)** Let present age of Nuri be x years and Sonu’s present age bey years.

**A.T.Q.**

1st Condition :

2nd Condition :

Subtractomg equation (ii) from equetion (i), we get

1st Condition :

2nd Condition :

Hence, present age of Nuri is 50 years and sonu’s present age is 20 years.

(iii) Let digit at unit place = x and digit at ten’s place = y.

Two digit number is lOy + x**A.T.Q.****1st Condition :**

x + y = 9**2nd Condition :**

9(10y + x) = 2(10k + y) ⇒ 90y + 9x = 20x + 2y

⇒ 88y – 11x = 0 ⇒ -11y + 88y = 0

⇒ -x + 8y = 0

Adding equestion (i) and (ii), we get

**(iv)** Let the number of notes of ₹ 50 = x and the number of notes of ₹ 100 = y**A.T.Q****1st Condition :**

50x + 100y = 2000

⇒ x + 2y = 40**2nd Condition :**

(v) Let, fixed charge for first 3 days be ₹ x and additional charge per day after 3 days be y.**A.T.Q.****1st Condition :** as per Saritha

x + 4y = 27**2nd Condition :** as per Susy

Putting y = 3 in equation (i),

x + 4(3) = 27 ⇒ x + 12 = 27 ⇒ x = 15

Hence, fixed charge is ₹ 15 and charge for each extra day is ₹ 3.