Question

The sum of the ages of father and his son is 65 years. After 5 years, the father's age will be twice the age of his son. Express this age as a pair of linear equations in two variables and hence find the present ages of father and his son.

Answer 1

Hint:In this question let the present age of the father be some x years and the age of his son be y years. Use the constraints of the questions to formulate two linear equations in two variables depicting the relationship between x and y. Then use the methods of substitution or elimination to solve the two linear two equations in two variables. This will help approaching the problem.

Complete step-by-step answer:
Let the present age of the father be x years.
And the age of his son by years.
Now it is given that the sum of the ages of father and his son is 65 years.
Now construct the linear equation according to above information we have,
Therefore, the present age of the father + present age of his son = 65 years.
Therefore, x + y = 65....................... (1)
Now it is also given that after 5 years the age of the father will be twice the age of his son.
So, after 5 years the age of the father = (x + 5) years.
And the age of his son after five years = (y + 5) years.
Now again construct the linear equation according to above information we have,
Age of the father after 5 years = 2(age of his son after 5 years).
Therefore, x + 5 = 2(y + 5)
 $ \Rightarrow x + 5 = 2y + 10 $
 $ \Rightarrow x = 2y + 5 $ .................. (2)
So equation (1) and (2) represents the pair of linear equations in two variables x and y.
Now substitute the value of x from equation (2) in equation (1) we have,
 $ \Rightarrow 2y + 5 + y = 65 $
Now simplify this we have,
  $ \Rightarrow 3y = 65 - 5 = 60 $
 $ \Rightarrow y = \dfrac{{60}}{3} = 20 $ Years.
Now from equation (2) we have,
 $ \Rightarrow x = 2\left( {20} \right) + 5 = 40 + 5 = 45 $ Years.
So the present age of the father is 45 years and his son is 20 years.
So this is the required answer.

Note: In the above problem we have used the method of substitution that is first relationship of one variable with the other is taken out and then this is put into other equation in order to make a single equation in one variable, solving it provides the value of this variable and then the relationship obtained is used to find the other variable, however there can be another method that is the method of elimination , in this we simply make the coefficients of any one variable same for both the equations and then by application of some basic arithmetic operations like addition/subtraction we eliminate this variable and thus an equation is obtained in a single variable only, solve it to get the value of the variable. The value of the other previously eliminated variable can be obtained by simply equating any one of the two equations given.