CBSE Class 10 Mathematics Chapter 18: Linear Equations in Two Variables NCERT Solutions

Solution:

Let the present age of Aftab be represented by x years, and the present age of his daughter be represented by y years.

According to the first condition (Seven years ago):

Aftab's age seven years ago was x - 7.

His daughter's age seven years ago was y - 7.

The problem states that Aftab was seven times as old as his daughter then. So, we can write the equation:

$$x\; -\; 7\; =\; 7(y\; -\; 7)$$

Expanding this equation:

$$x\; -\; 7\; =\; 7y\; -\; 49$$

Rearranging the terms to form a linear equation:

$$x\; -\; 7y\; =\; -49\; +\; 7$$

$$x\; -\; 7y\; =\; -42\; \backslash quad\; \backslash quad\; (1)$$

According to the second condition (Three years from now):

Aftab's age three years from now will be x + 3.

His daughter's age three years from now will be y + 3.

The problem states that Aftab will be three times as old as his daughter then. So, we can write the equation:

$$x\; +\; 3\; =\; 3(y\; +\; 3)$$

Expanding this equation:

$$x\; +\; 3\; =\; 3y\; +\; 9$$

Rearranging the terms to form a linear equation:

$$x\; -\; 3y\; =\; 9\; -\; 3$$

$$x\; -\; 3y\; =\; 6\; \backslash quad\; \backslash quad\; (2)$$

Algebraic Representation:

The situation can be represented algebraically by the following pair of linear equations:

$x\; -\; 7y\; =\; -42$

$x\; -\; 3y\; =\; 6$

Graphical Representation:

To represent these equations graphically, we need to find at least two points that satisfy each equation.

For equation (1): $x\; -\; 7y\; =\; -42$ or $x\; =\; 7y\; -\; 42$

If we choose $y\; =\; 5$, then $x\; =\; 7(5)\; -\; 42\; =\; 35\; -\; 42\; =\; -7$. So, one point is (-7, 5).

If we choose $y\; =\; 6$, then $x\; =\; 7(6)\; -\; 42\; =\; 42\; -\; 42\; =\; 0$. So, another point is (0, 6).

If we choose $y\; =\; 7$, then $x\; =\; 7(7)\; -\; 42\; =\; 49\; -\; 42\; =\; 7$. So, another point is (7, 7).

For equation (2): $x\; -\; 3y\; =\; 6$ or $x\; =\; 3y\; +\; 6$

If we choose $y\; =\; 0$, then $x\; =\; 3(0)\; +\; 6\; =\; 6$. So, one point is (6, 0).

If we choose $y\; =\; -1$, then $x\; =\; 3(-1)\; +\; 6\; =\; -3\; +\; 6\; =\; 3$. So, another point is (3, -1).

If we choose $y\; =\; -2$, then $x\; =\; 3(-2)\; +\; 6\; =\; -6\; +\; 6\; =\; 0$. So, another point is (0, -2).

Now, we plot these points on a graph paper and draw the lines connecting them. The intersection point of these two lines will give the solution to the system of equations.

Plotting the points (-7, 5), (0, 6), and (7, 7) for the first equation, and (6, 0), (3, -1), and (0, -2) for the second equation, we can draw the lines. The intersection point is found to be (42, 12).

Verification:

For equation (1): $42\; -\; 7(12)\; =\; 42\; -\; 84\; =\; -42$ (Correct)

For equation (2): $42\; -\; 3(12)\; =\; 42\; -\; 36\; =\; 6$ (Correct)

Thus, Aftab's present age is 42 years and his daughter's present age is 12 years.