Linear Equations in Two Variables – Algebraic and Graphical Representation

Video Explanation

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.” Represent this situation algebraically and graphically.

Solution

Step 1: Define the Variables

Let

\[ x = \text{Aftab’s present age (in years)} \]

\[ y = \text{Daughter’s present age (in years)} \]

Step 2: Form the Linear Equations

First Condition

Seven years ago:

\[ x – 7 = 7(y – 7) \]

\[ x – 7 = 7y – 49 \]

\[ x – 7y + 42 = 0 \]

Second Condition

Three years from now:

\[ x + 3 = 3(y + 3) \]

\[ x + 3 = 3y + 9 \]

\[ x – 3y – 6 = 0 \]

Step 3: Algebraic Representation

The required pair of linear equations is:

\[ \boxed{x – 7y + 42 = 0} \]

\[ \boxed{x – 3y – 6 = 0} \]

Step 4: Graphical Representation

Equation (1): \(x – 7y + 42 = 0\)

\[ x = 7y – 42 \]

y	x
7	7
8	14

Equation (2): \(x – 3y – 6 = 0\)

\[ x = 3y + 6 \]

y	x
5	21
6	24

Plot the points of both equations on the same graph. The two straight lines intersect at the point:

\[ (21,\, 3) \]

Conclusion

From the point of intersection:

\[ x = 21,\quad y = 3 \]

So,

Aftab’s present age = 21 years
Daughter’s present age = 3 years

Thus, the situation is correctly represented both algebraically and graphically.