Finding Two Numbers Given Their Sum and Relationship

In mathematical problem-solving, it is common to encounter scenarios where the sum of two numbers and a relationship between them are provided, and we need to find the actual values of these numbers. This article will walk through the process of solving such problems using algebraic equations. We will discuss a detailed step-by-step approach to ensure clarity and understanding.

Example Problem: Finding the Two Numbers

Let's consider the following problem: The sum of two numbers is 70. If three times the smaller number is subtracted from the larger number, the result is 6. How do we find the two numbers?

Step-by-Step Solution

Let's denote the larger number as L (Larger) and the smaller number as S (Smaller).

Based on the problem statement, we can write the following equations:

The sum of the two numbers is 70: L S 70 The larger number minus three times the smaller number is 6: L - 3S 6

Now, let's solve these equations step-by-step:

From the first equation, we can express L in terms of S: L 70 - S Substitute the expression for L into the second equation: 70 - S - 3S 6 Simplify the equation: 70 - 4S 6 Solve for S: 4S 64 S 16 Substitute the value of S back into the first equation to find L: L 70 - 16 54 So, the larger number is 54 and the smaller number is 16.

Verification

To verify our solution, we can check the given conditions:

The sum of the numbers: 54 16 70 Three times the smaller number subtracted from the larger number: 54 - 3 * 16 54 - 48 6

Conclusion

The two numbers are 54 and 16. This method, involving setting up and solving a system of linear equations, is a powerful tool for solving such problems.

Additional Examples

Let's look at another example to familiarize ourselves with this process:

Let y be the smaller number. Then the larger number is 70 - y. Setting up the second equation based on the problem statement: 70 - y - 3y 6. Simplify and solve for y: 70 - 4y 6 → 4y 64 → y 16. Therefore, the larger number is 70 - 16 54. Verification: 54 16 70 and 54 - 3 * 16 6.

Properties of the Problem

This type of problem involves basic arithmetic and algebraic manipulation. The key is to express one variable in terms of the other and then use the given relationship to formulate an equation.

Note on Problem-Solving Techniques

Understanding and practicing these types of problems helps in developing strong foundational skills in algebra and problem-solving. It is also useful to apply these techniques in real-world scenarios, such as budgeting, engineering, and various scientific applications.

Summary

By following the step-by-step process, we can easily solve the problem of finding two numbers given their sum and a relationship between them. This method can be applied to a wide range of similar problems in mathematics and practical applications.