Solving the Digit-Related Algebraic Problem: Deriving the Number from Given Digits

Number theory concepts often incorporate digits and their relationships. One digit-related problem involves figuring out a two-digit number based on the sum of its digits and a condition related to reversing the digits and subtraction. Let's solve this problem step by step, using algebraic equations to find the solution.

Problem Statement

Given a two-digit number, where:

The sum of the digits is 9. If 27 is subtracted from the number, the digits are reversed.

We need to determine the two-digit number.

Solution Approach

We will solve this problem by assigning the tens digit as (x) and the units digit as (y). The two-digit number can be expressed as (1 y).

Step 1: Sum of Digits

According to the problem, the sum of the digits is 9:

[begin{align*} x y 9 text{(Equation 1)} end{align*}]

Step 2: Digit Reversal and Subtraction

It is also given that if 27 is subtracted from the number, the digits are reversed. Therefore:

[begin{align*} 1 y - 27 10y x text{(Equation 2)} end{align*}]

Let's simplify Equation 2:

[begin{align*} 1 y - 27 10y x 9x - 9y 27 x - y 3 text{(Equation 3)} end{align*}]

Step 3: Solving the System of Equations

We have the following system of linear equations:

[begin{align*} x y 9 text{(Equation 1)} x - y 3 text{(Equation 3)} end{align*}]

Add the two equations to eliminate (y):

[begin{align*} (x y) (x - y) 9 3 2x 12 x 6 end{align*}]

Substitute (x 6) into Equation 1 to find (y):

[begin{align*} 6 y 9 y 3 end{align*}]

Therefore, the number is (1 y 10(6) 3 63).

Verification

To verify the solution, check if the sum of the digits is 9 and if reversing the digits and subtracting 27 results in the original number:

[begin{align*} 6 3 9 63 - 27 36 text{ (reversed to 36, which is 63)} end{align*}]

Both conditions are satisfied, confirming that the number is indeed 63.

Conclusion

The problem involves solving a system of linear equations derived from the given conditions. By assigning algebraic variables and manipulating the equations, we determined that the number is 63. This step-by-step approach ensures a clear and logical solution to the digit-related problem.