Solving the Mystery of Numbers with a Sum and Difference

Imagine you are presented with a puzzle: the sum of two numbers is 10, and their difference is 2. How would you solve this mystery? In this article, we will walk through various methods to find the numbers in question. We'll explore a simple one-variable solution, step-by-step algebraic equations, and the concept of mean in solving such problems.

One-Variable Method: Simplifying the Puzzle

Let's start with a simpler method by using one variable. If x is one number, then the other number is x - 2. Let's solve it step-by-step:

Set up the equations based on the given conditions: The sum of the numbers is 10: x (x - 2) 10 The difference between the numbers is 2: x - (x - 2) 2 Simplify and solve the equation: 2x - 2 10 2x 12 x 6 Now, find the other number: x - 2 6 - 2 4 Therefore, the numbers are 6 and 4.

By breaking down the problem, we can easily find the solution with minimal steps.

Step-by-Step Algebraic Equations: A Detailed Analysis

Let's use the traditional method with two variables. Let x and y be the numbers. We know:

x y 10 x - y 2

Let's solve these equations step-by-step:

From the second equation, isolate one of the variables: x - y 2 x y 2 Substitute x y 2 into the first equation: (y 2) y 10 2y 2 10 2y 8 y 4 Now, substitute y 4 back into the equation for x: x 4 2 6 Thus, the numbers are 6 and 4.

Mean and Difference: A Deeper Insight

Another approach to solving such problems is to use the concept of mean. If the sum of two numbers is 10, then their mean is half of 10, which is 5. The difference given is 2. Therefore, the two numbers must be 5 1 and 5 - 1, which are 6 and 4.

Let's verify:

The sum of 6 and 4 is 10. The difference of 6 and 4 is 2.

With this method, you can quickly determine the numbers without complex algebraic manipulations.

Conclusion

By using one variable, step-by-step algebraic equations, or the concept of mean, we can solve the puzzle of two numbers with a given sum and difference. The numbers, in this case, are 6 and 4.

Understanding these methods can help you solve similar problems in algebra. Practice these techniques to become more proficient in solving equations and number puzzles.