The difference between the two numbers is 4. If the sum of five times the smaller number and the bigger number is 58, find the numbers.

Hint: We can assume the bigger number and the smaller number as 2 variables. From the 1st statement, we can form a linear equation in 2 variables. From the 2nd statement, we get another equation. We can solve for the value of the 2 variables to get the required number.

Complete step by step answer:

Let us take x as the bigger number and y as the smaller number.
It is given that the difference between the numbers is 4. So, we can write it as an equation.
 $ \Rightarrow x - y = 4$ .. (1)
We are also given that sum of 5 times the smaller number and the bigger number is 58. We can form an equation as follows.
 $ \Rightarrow x + 5y = 58$ … (2)
Now we have 2 equations in x and y.
We can obtain the value of x and y by solving equation (1) and (2)
For solving, we can subtract equation (1) from (2)
 $
  {\text{ }}x + 5y = 58 \\
  \underline {\left( - \right)x - y = {\text{ }}4} \\
  {\text{ }}0x + 6y = 54 \\
 $
 $ \Rightarrow 6y = 54$
On dividing throughout with 6, we get,
 $ \Rightarrow y = \dfrac{{54}}{6}$
By further calculations, we get,
 $ \Rightarrow y = 9$
To get the value of x, we can substitute the value of y in equation (1)
 $ \Rightarrow x - 9 = 4$
On rearranging, we get,
 $ \Rightarrow x = 9 + 4$
On doing further calculations, we get,
 $ \Rightarrow x = 13$
Therefore, the value of x and y are 13 and 9 respectively.
So, the numbers are 13 and 9.

Note: In this problem, we made mathematical equations from the statements and solved the equation to get the required answer. This concept of making mathematical equations from statements is known as mathematical modeling. We formed 2 linear equations as we have 2 variables. We cannot solve a system of the equation if the number or equation is less than the number of variables. We can always check our answers by substituting the values of the variables back into the equations. We must read the given statements carefully before forming equations as it is the most important step in solving this type of problem.
An alternate method to solve the linear equations is by substitution.
We have equations,
 $ \Rightarrow x - y = 4$ … (1) and
 $ \Rightarrow x + 5y = 58$ … (2)
We can write equation (1) in terms of x.
 $ \Rightarrow x = 4 + y$ … (3)
On substituting this equation for x in equation (2), we get,
 $ \Rightarrow x + 5y = 58$
 $ \Rightarrow 4 + y + 5y = 58$
On rearranging, we get,
 $ \Rightarrow 6y = 54$
On dividing throughout with 6, we get,
 $ \Rightarrow y = \dfrac{{54}}{6}$
 $ \Rightarrow y = 9$
On substituting the value of y in equation (3), we get,
 $ \Rightarrow x = 4 + 9$
 $ \Rightarrow x = 13$
So, the numbers are 13 and 9.