NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 5

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 by the method of substitution.

Given:

Let the present age of Aftab be $ x $ years.
Let the present age of his daughter be $ y $ years.
Seven years ago, Aftab's age was $ x - 7 $ years, daughter's age was $ y - 7 $ years.
Seven years ago, Aftab was seven times as old as his daughter: $ x - 7 = 7(y - 7) $.
Three years from now, Aftab's age will be $ x + 3 $ years, daughter's age will be $ y + 3 $ years.
Three years from now, Aftab will be three times as old as his daughter: $ x + 3 = 3(y + 3) $.

To Find:

The present ages of Aftab ($ x $) and his daughter ($ y $).
Algebraic representation of the situation.
Graphical representation by the method of substitution.

Formula:

Linear equations in two variables: $ ax + by = c $
Method of substitution: Solve one equation for one variable and substitute in the other.

Solution:

Step 1: Form the equations based on the given conditions.

$$ \begin{align*} &\text{Seven years ago:} \\ &\quad x - 7 = 7(y - 7) \\ &\text{Three years from now:} \\ &\quad x + 3 = 3(y + 3) \end{align*} $$

Step 2: Simplify both equations to standard form.

$$ \begin{align*} x - 7 &= 7y - 49 \\ x - 7y &= -42 \quad \text{(Equation 1)} \\ \\ x + 3 &= 3y + 9 \\ x - 3y &= 6 \quad \text{(Equation 2)} \end{align*} $$

Step 3: Express $ x $ in terms of $ y $ using Equation 2.

$$ x - 3y = 6 \\ \Rightarrow x = 6 + 3y $$

Step 4: Substitute the value of $ x $ in Equation 1 and solve for $ y $.

$$ \begin{align*} x - 7y &= -42 \\ (6 + 3y) - 7y &= -42 \\ 6 + 3y - 7y &= -42 \\ 6 - 4y &= -42 \\ -4y &= -42 - 6 \\ -4y &= -48 \\ y &= \frac{-48}{-4} \\ y &= 12 \end{align*} $$

Step 5: Substitute $ y = 12 $ in the expression for $ x $ to find $ x $.

$$ x = 6 + 3y = 6 + 3 \times 12 = 6 + 36 = 42 $$

Step 6: State the solution.

Aftab's present age ($ x $) = 42 years
Daughter's present age ($ y $) = 12 years

Step 7: Represent the equations graphically.

Rewrite the equations in the form $ x = my + c $:

$$ \begin{align*} x - 7y &= -42 \implies x = 7y - 42 \\ x - 3y &= 6 \implies x = 3y + 6 \end{align*} $$

Choose values of $ y $ to get corresponding values of $ x $:

For $ x - 7y = -42 $:

$ y $	$ x $
0	-42
6	0
12	42

For $ x - 3y = 6 $:

$ y $	$ x $
0	6
6	24
12	42

The two lines intersect at $ (42, 12) $, which is the solution.

Result:

Aftab's present age = 42 years
Daughter's present age = 12 years
Algebraic equations:
- $ x - 7y = -42 $
- $ x - 3y = 6 $
Graphically, the lines $ x - 7y = -42 $ and $ x - 3y = 6 $ intersect at $ (42, 12) $.

Next question solution:

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 6

Explore more in Pair of Linear Equations:

Click this link to explore more NCERT Class X Chapter 3 Pair of Linear Equations

Explore more:

Click this link to explore more NCERT Class X chapter solutions

Search This Blog

Kaliyuga Ekalavya