Question

One number is greater than thrice the other number by 2. If 4 times the smaller number exceeds the greater by 5, find the numbers.

Answer 1

Hint: Let the two numbers be $x$ and $y$. Express the given information in the form of equations and solve them simultaneously. Once you find the value of either $x$ or $y$, substitute it in one of the equations to find the value of the other variable.

Complete step by step answer:
Using the given information, we will express the relation between the numbers in the form of an equation using some variables.
As we need to find only the greater and the smaller numbers, we will need only two variables.
Let the greater of the two numbers be denoted by $x$ and the smaller number be denoted by $y$
The first relation is: One number is greater than thrice the other number by 2
This relation means that the greater number = 3 times the smaller number + 2.
Thus, we get our first equation $x = 3y + 2...........(1)$
The second relation is: 4 times the smaller number exceeds the greater by 5
This means that the greater number = 4 times the smaller number - 5
Thus, the second equation will be $x = 4y - 5...........(2)$
Now, we have a system of 2 linear equations in 2 variables.
$x = 3y + 2...........(1)$
$x = 4y - 5...........(2)$
We need to solve these equations to find the required answer as we are looking for the values of $x$ and $y$
To solve this system of equations, we rearrange the terms in the equations such that the variables will be on the left hand side and the constants will be on the right hand side.
$x - 3y = 2...........(3)$
$x - 4y = - 5...........(4)$
We can see that the variable $x$ has the same coefficient in the equations (3) and (4).
Therefore, we will subtract (4) from (3).
$
  x - 3y - (x - 4y) = 2 - ( - 5) \\
   \Rightarrow x - 3y - x + 4y = 2 + 5 \\
   \Rightarrow x - x + 4y - 3y = 7 \\
   \Rightarrow y = 7 \\
 $
We have found the smaller of the two numbers which is 7.
To find the greater number we can substitute $y = 7$ in any of the equations (1), (2), (3), or (4).
Let’s use equation (1).
$y = 7 \Rightarrow x = 3y + 2 = 3(7) + 2 = 21 + 2 = 23$
Thus, the greater number is 23.

Hence the required numbers are 7 and 23.

Note: Alternate method of solving the system of linear equations
$x = 3y + 2...........(1)$
$x = 4y - 5...........(2)$
Express y in term of x in (2) and substitute the value in (1)
$x = 3y + 2...........(1)$
$
  x = 4y - 5...........(2) \\
   \Rightarrow x + 5 = 4y \\
   \Rightarrow \dfrac{{x + 5}}{4} = y \\
   \Rightarrow y = \dfrac{{x + 5}}{4}.......(3) \\
 $
Now, substitute (3) in (1)
\[
  x = 3y + 2 = 3\dfrac{{(x + 5)}}{4} + 2 \\
   \Rightarrow x = \dfrac{{3(x + 5) + 2 \times 4}}{4} = \dfrac{{3x + 15 + 8}}{4} = \dfrac{{3x + 23}}{4} \\
   \Rightarrow x = \dfrac{{3x + 23}}{4} \\
   \Rightarrow 4x = 3x + 23 \\
   \Rightarrow 4x - 3x = 23 \\
   \Rightarrow x = 23 \\
 \]
Now, substitute\[x = 23\]in (3)
$
  y = \dfrac{{x + 5}}{4} \\
   \Rightarrow y = \dfrac{{23 + 5}}{4} \\
   \Rightarrow y = \dfrac{{28}}{4} \\
   \Rightarrow y = 7 \\
 $
As can be seen above, both the methods give us the same values of$x$ and $y$