What is the age of a child whose age is two thirds of his elder brother, after $5$ years he will be four fifths.

Hint: It is the question of linear equations in two variables. We have to make two equations according to the situation given in the question. Then we can solve it by substitution method. Don’t get confused in their ages after five years as it is incremented in both the cases by $5$.

Complete step-by-step solution:
Let the age of the child be x.
Also, let the age of the elder brother be y.
Now, according the question
$x = \dfrac{{2y}}{3} - - - - \left( 1 \right)$
Also,
$ \Rightarrow \left( {x + 5} \right) = \dfrac{4}{5}\left( {y + 5} \right)$
On cross-multiplication, we get
$ \Rightarrow 5\left( {x + 5} \right) = 4\left( {y + 5} \right)$
On multiplication, we get
$ \Rightarrow 5x + 25 = 4y + 20$
On transposing, we get
$ \Rightarrow 4y - 5x = 25 - 20$
$ \Rightarrow 4y - 5x = 5$
Now, from equation $1$
$ \Rightarrow 4y - 5 \times \dfrac{{2y}}{3} = 5$
Taking LCM
$ \Rightarrow \dfrac{{12y - 10y}}{3} = 5$
Using cross-multiplication
$ \Rightarrow 2y = 5 \times 3$
$ \Rightarrow y = \dfrac{{15}}{2}$
$ \Rightarrow y = 7.5\,\,years$
Now, using equation $1$
$ \Rightarrow x = \dfrac{2}{3} \times \dfrac{{15}}{2}$
$ \Rightarrow x = 5\,\,years$
Therefore, the age of a child is $5$ years.

Note: linear equations in two variables deals with finding the solutions for pairs of such equations which have two variables present in it. These solutions can also be represented in graphs. It is an equation that is put in the form of \[ax{\text{ }} + by{\text{ }} + c = 0\] where, a, b and c are the real numbers and x and y are the variables. We can say that a and b are not equal to zero. The solution of linear equations is solved when the same number is added to both the sides and also when both the sides are multiplied and divided by the same number. Linear equations in two variables are expressed as x and y.