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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.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

Solution :-
  Let us denote age of Aftab x and age of his daughter y. 
  Then the algebraic representation is given by the following equations:

  x – 7 = 7 (y – 7)
  x - 7 = 7y - 49 
  x - 7y = -49 + 7
  x - 7y = -42  .......(i)
  
  x + 3 = 3 (y + 3) 
  x + 3 = 3y + 9  
  x - 3y = 9 - 3  
  x - 3y = 6 ......(ii)

  Lets represent these equations in graphically: 
  x - 7y = -42  .......(i)
x07
y67
x - 3y = 6 ......(ii)
x06
y-20
Plot the points A(0, 6), B(-42, 0) and P(0, -2), Q(6, 0), corresponding to the solutions in Table. Now draw the lines AB and PQ, representing the equations x - 7y = -42 and x - 3y = 6, as shown in following Figure:

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 x and y be the bats and balls.
  3x + 6y = 3900 
   x + 2y = 1300 .......(i)

   x + 2y = 1300 ........(ii)
  
  Lets represent these equations in graphically: 
  x + 2y = 1300 .......(i)
x01300
y6500
x + 2y = 1300 ........(ii)
x01300
y6500
Plot the points A(0, 650), B(1300, 0) and P(0, 650), Q(1300, 0), corresponding to the solutions in Table. Now draw the lines AB and PQ, representing the equations x + 2y = 1300 and x + 2y = 1300, as shown in following Figure: But, this is the same as Equation (i). Hence the lines represented by Equations (i) and (ii) are coincident. Therefore, Equations (i) and (ii) have infinitely many solutions.

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

Solution :-
  Let us denote apples by x and grapes by y. 
  The algebraic representation is given by the following equations:

  2x + y = 160  ...........(i) 

  4x + 2y = 300  
  2x + y = 150...........(ii)
  
  Lets represent these equations in graphically: 
  2x + y = 160  ...........(i) 
x5060
y6040
2x + y = 150...........(ii)
x7075
y100
Plot the points A(0, 160), B(80, 0) and P(0, 150), Q(75, 0), corresponding to the solutions in Table. Now draw the lines AB and PQ, representing the equations 2x + y = 160 and 2x + y = 150, as shown in following Figure:
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