Form a pair of linear equations for the following problems and find their solution by substitution method.

**(i)** The difference between two numbers is 26 and one number is three times the other. Find them.

**Solution:**

Let first number be x

Let second number be y

According to given conditions, we have

(assuming x>y) **(1)**

(x>y) ** (2)**

Putting equation **(2)** in **(1)**, we get

Putting value of **y** in equation **(2)**, we get

Therefore, two numbers are **13 and 39.**

**(ii) **The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

**Solution:**

Let smaller angle

Let larger angle

According to given conditions, we have

** (1)**

Also, (**Sum of supplementary angles**) ** (2)**

Putting **(1)** in equation **(2)**, we get

Putting value of **x** in equation** (1)**, we get

Therefore, two angles are and

**(iii) **The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

**Solution:**

Let cost of each bat = Rs x

Let cost of each ball = Rs y

According to given conditions, we have

**(1)**

And, ** (2)**

Using equation **(1)**, we can say that

Putting this in equation **(2)**, we get

Putting value of y in **(2)**, we get

Therefore, cost of each bat = Rs 500

Cost of each ball = Rs 50

**(iv) **The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

**Solution:**

Let fixed charge = Rs x

Let charge for every km = Rs y

According to given conditions, we have

**(1)**

**(2)**

Using equation **(1)**, we can say that

Putting this in equation **(2)**, we get

Putting value of **y** in equation **(1)**, we get

Therefore, fixed charge = Rs 5

Charge per km = Rs 10

To travel distance of 25 Km, person will have to pay = Rs (x+25y) = Rs (5+25(10)) = Rs(5+250) = Rs 255

**(v) **A fraction becomes , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and denominator it becomes . Find the fraction.

**Solution:**

Let numerator = x

Let denominator = y

According to given conditions, we have

**(1)**

** (2)**

Using equation **(1)**, we can say that

Putting value of **x** in equation **(2)**, we get

Putting value of **y** in **(1)**, we get

Therefore, fraction

**(vi)** Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages?

**Solution:**

Let present age of Jacob = x years

Let present age of Jacob’s son = y years

**According to given conditions, we have**

**(1)**

And, **(2)**

From equation **(1)**, we can say that

Putting value of **x** in equation **(2)** we get

**years**

Putting value of **y** in equation **(1)**, we get

years

Therefore, present age of Jacob = 40 years

And, present age of Jacob’s son = 10 years