# Linear Equations

## NCERT Exercise 3.1

Question 1: Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Represent this situation algebraically and graphically.

Solution: Let us assume Aftab’s current age = x and his daughter’s current age = y

Seven years ago: Aftab’s age = x - 7 and daughter’s age = y - 7

As per question;

x – 7 = 7(y – 7)
Or, x – 7 = 7y – 49
Or, x = 7y – 49 + 7
Or, x = 7y – 42
Or, 7y – x – 42 = 0 ……(1)

This equation gives following values for x and y

 x - 35 - 28 - 21 - 14 - 7 0 7 14 21 28 35 42 49 y 1 2 3 4 5 6 7 8 9 10 11 12 13

Three years from now:

Aftab’s age = x + 3 and daughter’s age = y + 3

As per question;

x + 3 = 3(y + 3)
Or, x + 3 = 3y + 9
Or, x = 3y + 9 – 3
Or, x = 3y + 6
Or, 3y – x + 6 = 0 ……..(2)

This equation gives following values for x and y:

 x 9 12 15 18 21 24 27 30 33 36 39 42 45 y 1 2 3 4 5 6 7 8 9 10 11 12 13

The following graph is plotted for the given pair of linear equations: Daughter’s age = 12 years and Aftab’s age = 42 years

Question 2: The coach of a cricket team buys 3 bats and 6 balls for Rs. 3900. Later, she buys another bat and 3 more balls of the same kind for Rs. 1300. Represent this situation algebraically and geometrically.

Solution: Let us assume that price of one bat = x and price of one ball = y. Following equations can be written as per the question:

3x + 6y = 3900
Or, x + 2y = 1300 ……..(1)

This equation will give following values for x and y

 x 1500 1400 1300 1200 1100 y - 100 - 50 0 50 100

x + 3y = 1300 ……….(2)

This equation will give following values for x and y

 x 1600 1450 1300 1150 1000 y - 100 - 50 0 50 100

The following graph is plotted for the given pair of linear equations. Price of one bat = Rs. 1300 and Price of one ball = zero

Question 3: The cost of 2 kg apples and 1 kg grapes was found to be Rs. 160. After a month, the cost of 4 kg apples and 2 kg grapes is Rs. 300. Represent this situation algebraically and graphically.

Solution: Let us assume that cost of 1 kg apple = x and cost of 1 kg grapes = y. Following equations can be written as per the question.

2x + y = 160 ……..(1)

This equation gives following values for x and y

 x 70 60 50 40 y 20 40 60 80

4x + 2y = 300
Or, 2x + y = 150 ……..(2)

This equation will give following values for x and y:

 x 65 55 45 35 y 20 40 60 80

The following graph is plotted for the given pair of linear equations. Since we get parallel lines so there will be no solution for this pair of linear equations.