Five years ago, Adil was thrice as old as Bharat. Ten years later A...

Question

Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. To know the present ages of Adil and Bharat :
(i) form the linear equations representing the above information.
(ii) show that the system of equations is consistent with unique solution.
(iii) find the present ages of Adil and Bharat.

Accepted Answer

Let the present age of Adil be $x$ years and that of Bharat be $y$ years.

(i) Forming the equations
Step 1: Five years ago : Adil $= x - 5$, Bharat $= y - 5$.
$$x - 5 = 3(y - 5) \Rightarrow x - 3y + 10 = 0 \qquad \ldots (1)$$
Step 2: Ten years later : Adil $= x + 10$, Bharat $= y + 10$.
$$x + 10 = 2(y + 10) \Rightarrow x - 2y - 10 = 0 \qquad \ldots (2)$$

(ii) Consistency
Step 3: Here $a_1 = 1,\ b_1 = -3,\ c_1 = 10$ and $a_2 = 1,\ b_2 = -2,\ c_2 = -10$.
$$\frac{a_1}{a_2} = \frac{1}{1} = 1,\qquad \frac{b_1}{b_2} = \frac{-3}{-2} = \frac{3}{2}$$
Step 4: Since $\dfrac{a_1}{a_2} 
eq \dfrac{b_1}{b_2}$, the system is CONSISTENT and has a UNIQUE solution (the lines intersect at exactly one point).

(iii) Solving
Step 5: Subtract $(2)$ from $(1)$ :
$$(x - 3y + 10) - (x - 2y - 10) = 0 \Rightarrow -y + 20 = 0 \Rightarrow y = 20$$
Step 6: Substitute $y = 20$ in $(2)$ :
$$x - 2(20) - 10 = 0 \Rightarrow x = 50$$
Hence the present age of Adil $= 50$ years and of Bharat $= 20$ years.
Check : $5$ years ago $45 = 3 	imes 15$ \checkmark ; $10$ years later $60 = 2 	imes 30$ \checkmark

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