Question

The sum of two numbers is 8. If their sum is four times their difference, find the numbers.

Answer 1

Hint- Here, we will proceed by assuming the two numbers as two different variables say x and y. Then, we will form two linear equations in these two variables and solve them with the help of the substitution method.

Complete step-by-step solution -
Let the two numbers be x and y where x > y
Given, sum of two numbers is 8 i.e.,
$
  x + y = 8 \\
  \Rightarrow x = 8 - y{\text{ }} \to {\text{(1)}} \\
 $
Also given that the sum of these two numbers is four times the difference between these two numbers
i.e., $
  x + y = 4\left( {x - y} \right) \\
   \Rightarrow x + y = 4x - 4y \\
   \Rightarrow 4x - x - 4y - y = 0 \\
   \Rightarrow 3x - 5y = 0 \\
 $
By substituting the value of x from equation (1) in the above equation, we get
\[
   \Rightarrow 3\left( {8 - y} \right) - 5y = 0 \\
   \Rightarrow 24 - 3y - 5y = 0 \\
   \Rightarrow 8y = 24 \\
   \Rightarrow y = 3 \\
 \]
By putting y = 3 in equation (1), we get
$\Rightarrow x = 8 - 3 = 5 $
Therefore, the two required numbers are 5 and 3.

Note- In this particular problem, we have assumed the number x greater than the number y. Here, if we would have assumed the number y greater than the number x then we would have got x = 3 and y = 5 as the result. Here, in order to solve the two linear equations in two variables we can also proceed by elimination method.