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Ravish tells his daughter Aarushi, “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”. If present ages of Aarushi and Ravish are $x$ and $y$ years respectively, represent this situation algebraically as well as graphically.
Given:
Ravish tells his daughter Aarushi, “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”.
The present ages of Aarushi and Ravish are $x$ and $y$ years respectively.
To do:
We have to represent the above situation algebraically and graphically.
Solution:
$x$ is the present age of Aarushi and $y$ is the present age of Ravish.
Age of Aarushi 7 years ago$=x-7$.
Age of Ravish 7 years ago$=y-7$.
Age of Aarushi 3 years later$=x+3$.
Age of Ravish 3 years later$=y+3$.
According to the question,
$x-7 = 7(y-7)$
$\Rightarrow x-7=7y-49$
$\Rightarrow x-7y+42= 0$.....(i)
$7y=x+42$
$y=\frac{x+42}{7}$
Also,
$x+3 = 3(y+3)$
$\Rightarrow x+3=3y+9$
$\Rightarrow x-3y-6=0$......(ii)
$3y=x-6$
$y=\frac{x-6}{3}$
To represent the above equations graphically we need at least two solutions for each of the equations.
For equation (i),
If $x=-7$ then $y=\frac{-7+42}{7}=\frac{35}{7}=5$
If $x=0$ then $y=\frac{0+42}{7}=6$
If $x=7$ then $y=\frac{7+42}{7}=\frac{49}{7}=7$
$x$ | $-7$ | $0$ | $7$ |
$y=\frac{x+42}{7}$ | $5$ | $6$ | $7$ |
For equation (ii),
If $x=0$ then $y=\frac{0-6}{3}=\frac{-6}{3}=-2$
If $y=0$ then $0=\frac{x-6}{3}$
$\Rightarrow x=6$
If $x=3$ then $y=\frac{3-6}{3}=\frac{-3}{3}=-1$
$x$ | $6$ | $3$ | $0$ |
$y=\frac{x-6}{3}$ | $0$ | $-1$ | $-2$ |
The above situation can be plotted graphically as below:
The line AC represents the equation $x-7y+42=0$ and the line PR represents the equation $x-3y-6=0$.