### Question 1. Solve the following pair of linear equations by the elimination method and the substitution method:

### (i) x + y = 5 and 2x – 3y = 4

**Solution:**

Here, the two given eqn. are as follows:

x + y = 5 ……….(I)

2x – 3y = 4 ………..(II)

ELIMINATION METHOD:Multiply equation (I) by 2, and then subtract (II) from it, we get

5y = 6

y = 6/5

Now putting y=6/5 in eqn. (I), we get

x + 6/5 = 5

x = (5–(6/5))

x = 19/5

SUBSTITUTION METHOD:From (I), we get

y=5–x…….(III)

Now substituting the value of y in eqn. (II), we get

2x – 3(5–x) = 4

2x – 15+3x = 4

5x = 4+15

x = 19/5

As, putting x = 19/5, in eqn. (III), we get

y = 5 – 19/5

y = 6/5

Hence, by elimination method and substitution method we get,

x = 19/5 and y = 6/5.

### (ii) 3x + 4y = 10 and 2x – 2y = 2

**Solution:**

Here, the two given eqn. are as follows:

3x + 4y = 10 ……….(I)

2x – 2y = 2 ………..(II)

ELIMINATION METHOD:Multiply equation (II) by 2, and then add it to (I), we get

7x = 14

x = 14/7

x = 2

Now putting x = 2 in eqn. (I), we get

3(2) + 4y = 10

4y = 10 – 6

y = 4/4

y = 1

SUBSTITUTION METHOD:From (II), we get

x = (2+2y)/2

x = y+1 …….(III)

Now substituting the value of x in eqn. (I), we get

3(y+1) + 4y = 10

3y + 3 + 4y = 10

7y = 10 – 3

y = 7/7

y = 1

As, putting y = 1, in eqn. (III), we get

x = 1+1

x = 2

Hence, by elimination method and substitution method we get,

x = 2 and y = 1.

### (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

**Solution:**

Here, the two given eqn. are as follows:

3x – 5y – 4 = 0

9x = 2y + 7

By rearranging we get,

3x – 5y = 4 ……….(I)

9x – 2y = 7 ………..(II)

ELIMINATION METHOD:Multiply equation (I) by 3, and then subtract (II) from it, we get

–13y = 5

y = -5/13

Now putting y = – 5/13 in eqn. (I), we get

3x – 5(– 5/13) = 4

3x = 4 – (25/13)

3x = 27/13

x = 9/13

SUBSTITUTION METHOD:From (I), we get

3x – 5y = 4

x = (4+5y)/3 …….(III)

Now substituting the value of x in eqn. (II), we get

9((4+5y)/3) – 2y = 7

3(4+5y) – 2y = 7

12+15y – 2y = 7

13y = – 5

y = – 5/13

As, putting y = – 5/13, in eqn. (III), we get

x=(4+5(– 5/13))/3

x = 9/13

Hence, by elimination method and substitution method we get,

x=9/13 and y=–5/13.

### (iv) x/2 + 2y/3 = -1 and x – y/3 = 3

**Solution:**

Here, the two given eqn. are as follows:

x/2 + 2y/3 = –1 ………..(A)

x – y/3 = 3 ……………….(B)

by rearranging (multiply (A) by 6 and multiply (B) by 3) we get,

3x + 4y = – 6 ……….(I)

3x – y = 9 ………..(II)

ELIMINATION METHOD:Subtract (II) from (I), we get

5y = – 15

y = – 3

Now putting y = -3 in eqn. (II), we get

3x – (– 3) = 9

3x = 9 – 3

x = 6/3

x = 2

SUBSTITUTION METHOD:From (II), we get

3x – y = 9

y = 3x – 9 …….(III)

Now substituting the value of y in eqn. (I), we get

3x + 4(3x – 9) = – 6

3x + 12x – 36 = – 6

15x = – 6 + 36

x = 30/15

x = 2

As, putting x = 2, in eqn. (III), we get

y = 3(2) – 9

y = – 3

Hence, by elimination method and substitution method we get,

x=2 and y=–3.

### Question 2. 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 ½ if we only add 1 to the denominator. What is the fraction?

**Solution:**

Let the fraction be p/q, where p is numerator and q is denominator.

Here, According to the given condition,

(p+1)/(q – 1) = 1 ………………..(A)

and,

p/(q+1) = 1/2 …………………..(B)

Solving (A), we get

(p+1) = q – 1

p–q =–2 ……………………….(I)Now, solving (B), we get

2p = (q+1)

2p–q = 1……………………….(II)When equation (I) is subtracted from equation (II) we get,

p = 3

Now putting p = 3 in eqn. (I), we get

3 – q = – 2

q = 3+2

q = 5

So, p = 3 and q = 5.

Hence, the fraction p/q is 3/5.

### (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?

**Solution:**

Let us assume, present age of Nuri is x

And present age of Sonu is y.

Here, According to the given condition, the equation formed will be as follows :

x – 5 = 3(y – 5)

x – 3y =–10 …………………………………..(I)Now,

x + 10 = 2(y +10)

x – 2y = 10 …………………………………….(II)Subtract eqn. (I) from (II), we get

y = 20

Now putting y = 20 in eqn. (II), we get

x – 2(20) = 10

x = 10+40

x = 50

Hence,

Age of Nuri is 50 years

Age of Sonu is 20 years.

### (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.

**Solution:**

Let the unit digit and tens digit of a number be x and y respectively.

Then, Number = 10y + x

And, reverse number = 10x + y

eg: 23

x = 3 and y = 2

So, 23 can be represented as = 10(2) + 3 = 23Here, According to the given condition

x + y = 9 …………………….(I)and,

9(10y + x) = 2(10x + y)

90y + 9x = 20x + 2y

88y = 11x

x = 8y

x–8y = 0 ………………………………………………………….. (II)Subtract eqn. (II) from (I) we get,

9y = 9

y = 1

Now putting y = 1 in eqn. (II), we get

x – 8(1) = 0

x = 8

Hence, the number is 10y + x

=10 × 1 + 8

Number = 18

### (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.

**Solution:**

Let the number of ₹ 50 notes be x and the number of ₹100 notes be y

Here, According to the given condition

x + y = 25 ……………………………….. (I)

50x + 100y = 2000 ………………………………(II)Divide (II) by 50 and then subtract (I) from it.

y = 15

Now putting y = 15 in eqn. (I), we get

x + 15 = 25

x = 10

Hence, Manna has 10 notes of ₹ 50 and 15 notes of ₹ 100.

### (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:**

Let the fixed charge for the first three days be ₹ x and,

The charge for each day extra be ₹ y.

Here, According to the given condition,

x + 4y = 27 …….…………………………. (I)

x + 2y = 21 ………………………………………….. (II)Subtract (II) from (I), we get

2y = 6

y = 3

Now putting y = 3 in eqn. (II), we get

x + 4(3) = 27

x = 27 – 12

x = 15

Hence, the fixed charge is ₹15

And the Charge per day is ₹ 3