# Pair of Linear Equations in Two Variables : Exercise - 3.1 (Mathematics NCERT Class 10th)

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**Q.1 Â Â Â 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". (Isn't this interesting)? Represent this situation algebraically and graphically.**

** Sol.**Â Â Â Â Let's denote the present age of daughter and her father Aftab as x years and y years respective. Then, algebraic representation is given by the following equations :

Â Â Â Â Â Â Â 7(x â€“ 7) = y â€“ 7

Â Â Â Â Â Â Â 7x â€“ 49 = y â€“ 7

Â Â Â Â Â Â Â 7x â€“ y = 42

Â Â Â Â Â Â Â and, 3(x + 3) = y + 3

Â Â Â Â Â Â Â 3x + 9 = y + 3

Â Â Â Â Â Â Â 3x â€“ y = â€“ 6

Â Â Â Â Â Â Â To obtain the equivalent graphical representation, we find two points on the line representing each equation. That is, Â we find two solutions of each equation.Â That is, we find two solutions of each equation.

Â Â Â Â Â Â These solutions are given below in the tables:

Â Â Â Â Â Â For 7x â€“ y = 42

Â Â Â Â Â Â For 3x â€“ y = â€“ 6

Â Â Â Â Â Â To represent these equations graphically, we plot the points A(6, 0) and B(5, â€“7) to get the graph of (i) and the points C(0, 6) and D(â€“2, 0) give the graph of (ii).

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

*Â Â Â Â Let us denote the cost of bat be Rs. x and one ball be Rs. y. Then, the algebraic representation is given by the following equations :*

**Sol.**Â Â Â Â Â Â Â 3x + 6y = 3900 x + 2y = 1300 ...(1)

Â Â Â Â Â Â Â and, x + 3y = 1300 ...(2)

Â Â Â Â Â Â Â To obtain the equivalent geometric representation, we find two points on the line representing each equation. That is, we find two solutions of each equation.

Â Â Â Â Â Â Â These solutions are given below in the table.

Â Â Â Â Â Â Â For x + 2y = 1300

Â Â Â Â Â Â Â For x + 3y = 1300

Â Â Â Â Â Â Â We plot the points A(0, 650), B(1300, 0) to obtain the geometric representation of x + 2y = 1300 and and B (1300, 0) to obtain the geometric representation of x + 3y = 1300.

Â Â Â Â Â Â Â We observe that these lines intersect at B (1300, 0).

**Q.3 Â Â Â The cost of 2 kg of apples and 1 kg of grapes on a day was found to be Rs. 160. After a months, the cost of 4 kg of apples and 2 kg of grapes is Rs. 300. Represent the situation algebraically and geometrically.
**

*Â Â Â Â Let us denote the cost of 1 kg of apple by Rs. x and cost of 1 kg grapes by Rs.y.*

**Sol.**Â Â Â Â Â Â Â Then the algebraic representation is given by the following equations :

Â Â Â Â Â Â Â 2x + y = 160 ...(1)

Â Â Â Â Â Â Â 4x + 2y = 300 ...(2)

Â Â Â Â Â Â Â 2x + y = 150

Â Â Â Â Â Â Â To find the equivalent geometric representation, we find two points on the line representing each equation. That is, we find two solutions of each equation.

Â Â Â Â Â Â Â 2x + y = 160Â

Â Â Â Â Â Â Â 2x + y = 150