Question

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.

Class 10 - Math - Pair of Linear Equations in Two Variables Page 44

Answer 1

Let present age of Aftab be x
And, present age of daughter is represented byy
Then Seven years ago,
Age of Aftab = x -7
Age of daughter = y-7
According to the question,
(x - 7)  = 7 (y – 7 )
x – 7 = 7 y – 49
x- 7y = - 49 + 7 
x – 7y = - 42 …(i)
x = 7y – 42 
Putting y = 5, 6 and 7, we get
x = 7 × 5 - 42 = 35 - 42 = - 7
x = 7 × 6 - 42 = 42 – 42 = 0
x = 7 × 7 – 42 = 49 – 42 = 7

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Three years from now ,
Age of Aftab = x +3
Age of daughter = y +3
According to the question,
(x + 3) = 3 (y + 3)
x + 3 = 3y + 9
x -3y = 9-3
x -3y = 6 …(ii)
x = 3y + 6 
Putting, y = -2,-1 and 0, we get
x = 3 × - 2 + 6 = -6 + 6 =0
x = 3 × - 1 + 6 = -3 + 6 = 3
x = 3 × 0 + 6 = 0 + 6 = 6


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Algebraic representation
From equation (i) and (ii)
x – 7y = – 42 …(i)
x - 3y = 6 …(ii)

Answer 2

Algebraically the situation will be represented as $(x-7)=7(y-7)$ and  $(x+3)=3(y+3)$.

Given:

Seven years ago, Aftab was seven times as old as his daughter was then. Also, three years from now, he shall be three times as old as his daughter will be.

To Find:

To represent this situation algebraically and graphically.

Solution:

Let the present age of Aftab is x years and the present age of Aftab's daughter be y years.

Seven years ago the age of Aftab was $(x-7)\,years$ and his daughter's age was $(y-7)\,years$.

Following the question, the equation for "seven years ago, Aftab was seven times as old as his daughter was then."

$(x-7)=7(y-7)\,\,\,...(1)$

Three years from the present, the age of Aftab will be $(x+3)\,years$ and his daughter's age will be $(y+3)\,years$.

Following the question, the equation for "three years from now, Aftab shall be three times as old as his daughter will be."

$(x+3)=3(y+3)\,\,\,...(2)$

To represent the equations (1) and (2) on the graph, a few points will be needed.

Substitute 0 for x into the equation (1) and find y.

$(0-7)=7(y-7)\\-7=7y-49\\7y=42\\y=6$

Substitute 0 for y into the equation (1) and find x.

$(x-7)=7(0-7)\\x-7=-49\\x=-42$

Two points are obtained as (0,6) and (-42,0) through which the line for equation (1) will pass.

Substitute 0 for x into the equation (2) and find y.

$(0+3)=3(y+3)\\3=3y+9\\3y=-6\\y=-2$

Substitute 0 for y into the equation (2) and find x.

$(x+3)=3(0+3)\\x+3=9\\x=6$

Two points are obtained as (0,-2) and (6,0) through which the line for equation (2) will pass.

Thus, algebraically the situation will be represented as $(x-7)=7(y-7)$ and  $(x+3)=3(y+3)$, and graphically it is represented as,

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