Question

One number is greater than thrice the other number by 2 and 4 times smaller number exceeds greater number by 5. Find the numbers.

Answer 1

Hint: Here, we will assume the greater and smaller number to be any variable. Then we will form two equations based on the given information.

Complete step-by-step answer:
Let \[x\] and \[y\] denote the greater number and smaller number respectively. We are given that one number is greater than thrice the other number by 2.
So, we have
\[ \Rightarrow x = 2 + 3y\]
Rewriting the above equation, we get
\[ \Rightarrow x - 3y = 2\] ………………………….. (1)
Then it is given that 4 times smaller number exceeds greater number by 5.
So, we have
\[ \Rightarrow 4y - x = 5\] ………………………(2)
Adding equation (1) and equation (2), we have
\[ \Rightarrow x - 3y + 4y - x = 2 + 5\]
Subtracting the like terms, we have
\[ \Rightarrow y = 7\]
Substituting the value of \[y = 7\], we have
\[ \Rightarrow x - 3(7) = 2\]
\[ \Rightarrow x - 21 = 2\]
Rewriting the equation, we have
\[ \Rightarrow x = 2 + 21\]
\[ \Rightarrow x = 23\]
Therefore, the numbers are \[23,7\].
Note: The above equation is solved using the method of elimination. In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
It can be solved by using the method of substitution. The method of substitution involves three steps which includes solving one equation for one of the variables, substituting this expression into the other equation and solve and then substituting the value into the original equation to find the corresponding variable