Logical reasoning is a powerful tool that can be used to solve complex mathematical problems efficiently. In this article, we will explore a practical example through which we can solve a system of equations without the need for abstract mathematical representations. We will demonstrate both logical reasoning and algebraic methods to find the solution to a simple yet instructive problem that many people encounter.

Introduction

Consider the following problem: Two numbers have a sum of 24 and a difference of 2. The question is, what are these two numbers?

Logical Reasoning Solution

This approach involves breaking down the problem into simpler components and using logical deductions to find the answer.

1. Let the two numbers be ( x ) and ( y ), where ( x > y ).

2. We know that their sum is 24: [ x y 24 ]

3. We also know that their difference is 2: [ x - y 2 ]

4. If we reduce the larger number ( x ) by 5, the two numbers will be equal. This is because reducing ( x ) by 5 will make it equal to ( y 5 ), which is the original value of ( y ) plus the constant difference of 2. The sum of the two new numbers will be 20 because the original sum of 25 is reduced by 5.

5. Now we have two equal numbers that add up to 20, so each number is 10. Since the original larger number ( x ) is 5 more than ( y ), we find that ( x 15 ) and ( y 10 ).

Algebraic Solution

If we prefer, we can solve this problem using algebraic methods. Here’s how:

1. Let the two numbers be ( x ) and ( y ), where ( x > y ).

2. We know that: [ x y 24 ] [ x - y 2 ]

3. By adding these two equations together:

[ (x y) (x - y) 24 2 ] [ 2x 26 ] [ x 13 ]

4. Substitute ( x 13 ) into the first equation:

[ 13 y 24 ] [ y 24 - 13 ] [ y 11 ]

Therefore, the two numbers are 13 and 11.

Conclusion

As demonstrated, logical reasoning and algebraic methods can both be effective in solving systems of equations. Logical reasoning can provide a clear and intuitive path to the solution, while algebraic methods provide a systematic and rigorous approach.

Using these methods, we conclude that the two numbers are 13 and 11. This solution can be verified by checking that 13 11 24 and 13 - 11 2, confirming the correctness of the solution.

Key Points:

Logical reasoning simplifies the problem by breaking it down into simpler components. Algebraic methods provide a structured approach using equations and operations. The solution to the problem is 13 and 11. Both methods lead to the same conclusion.