Question

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.

Answer 1

HINT: For solving the above question, we would be requiring the knowledge of solving the system of linear equations in two variables. In this question we would be using an elimination method. In elimination method, we first try to make the coefficient of any one variable of the two as equal and then subtract or add the new equations accordingly.
Then, we will get the equation which will be having only one variable.
Then we can solve the equation to get the value of that variable which is left and after getting the value of any one variable, we can plug in that value in any of the equations and then get the value of the other variable as well.

Complete step by step answer:
As mentioned in the question,
Let the age of Aftab be y and the age of his daughter be x.
Now 7 years ago, according to the question, we get
\[\begin{align}
  & 7\left( x-7 \right)=\left( y-7 \right) \\
 & 7x-y=42\ \ \ \ \ ...(a) \\
\end{align}\]
Now, after 3 years, according to the question, we get
\[\begin{align}
  & 3\left( x+3 \right)=\left( y+3 \right) \\
 & 3x-y=-6\ \ \ \ \ ...(b) \\
\end{align}\]
Now, on subtracting equations (a) and (b), we get
\[\begin{align}
  & \ \ \ \ 7x-y=42 \\
 & \dfrac{-(3x-y=-6)}{4x=48} \\
 & x=12 \\
\end{align}\]
Now, putting this value in (b), we get
\[\begin{align}
  & 3\times 12-y=-6 \\
 & y=42 \\
\end{align}\]
Hence, the present age of Aftab is 42 years and his daughter’s age is 12 years. \[\]

Graphically also, the solution is x=9, y=42.

NOTE:For questions in which there are more than 2 variables, in order to know whether the equations are solvable or whether we will be able to get the values of the variables by just counting the number of variables and number of the equations. If the number of equations and the number of variables involved in the question is equal then we can surely say that every variable will be having a unique value. If these numbers are not equal, then we do not comment on that.
There are two other methods of solving a 2 variable system of equations:-
substitution method
cross multiplication method