Solve the following 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 by the method of substitution.

Step 1: Define Variables

We assign variables to the unknown quantities. Let $x$ represent Aftab's current age and $y$ represent his daughter's current age. This is the first step in setting up any word problem algebraically.

Step 2: Formulate Equation from First Condition

The first condition states that seven years ago, Aftab was seven times as old as his daughter. Seven years ago, Aftab's age was $x-7$ and his daughter's age was $y-7$. We set up the equation based on this relationship and simplify it to form the first linear equation.

Step 3: Formulate Equation from Second Condition

The second condition states that three years from now, Aftab will be three times as old as his daughter. Three years from now, Aftab's age will be $x+3$ and his daughter's age will be $y+3$. We set up the equation based on this relationship and simplify it to form the second linear equation.

Step 4: Solve by Substitution Method

We will use the substitution method as requested. From equation (2), we express $x$ in terms of $y$. Then, we substitute this expression for $x$ into equation (1) to solve for $y$. This simplifies the system to a single variable equation.

Step 5: Find the Value of x

Now that we have the value of $y$, we substitute it back into equation (3) (or either of the original equations) to find the value of $x$. This gives us Aftab's present age.

Step 6: Graphing the First Equation

To graph the first equation, $x - 7y = -42$, we find a few points that satisfy the equation. We choose values for $y$ and calculate the corresponding $x$ values. These points will lie on the line representing the first equation.

Step 7: Graphing the Second Equation

Similarly, to graph the second equation, $x - 3y = 6$, we find a few points that satisfy this equation. We choose values for $y$ and calculate the corresponding $x$ values. These points will lie on the line representing the second equation. The intersection of the two lines will be the solution.

Step 8: State the Solution

The solution to the system of linear equations is $x=42$ and $y=12$. This means Aftab's present age is 42 years and his daughter's present age is 12 years. This point also represents the intersection of the two lines on the graph.

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