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:

1. Form the linear equations representing the above information.
2. Show that the system of equations is consistent with unique solution.
3. Find the present ages of Adil and Bharat.

Answer 1

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

(i) Forming the linear equations

Five years ago,

Adil’s age = x − 5 

Bharat’s age = y − 5 

According to the condition:

x − 5 = 3(y − 5) 

x − 5 = 3y − 15

x − 3y = −10    ...(1)

Ten years later:

Adil’s age = x + 10 

Bharat’s age = y + 10 

According to the condition:

x + 10 = 2(y + 10) 

x + 10 = 2y + 20 

x − 2y = 10   ...(2)

(ii) Comparing equations with a₁​x + b_1​y = c₁​ and a₂​x + b₂​y = c₂​:

a₁​ = 1, b₁ = −3, c₁ = −10

a₂​ = 1, b₂ = −2, c₂ = 10

`(a_1)/(a_2) = 1/1`

= 1

`(b_1)/(b_2) = (-3)/(-2)`

= `3/2`

Since `(a_1)/(a_2) ≠ (b_1)/(b_2)`

The system of equations is consistent and has a unique solution.

(iii) Subtracting equation (1) from equation (2):

(x − 2y) − (x − 3y) = 10 − (−10)

x − 2y − x + 3y = 10 + 10

y = 20

Putting y = 20 in equation (2):

x − 2(20) = 10

x − 40 = 10

x = 50

Present age of Adil = 50 years

Present age of Bharat = 20 years