# 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.