Linear Equations

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.