Solving Systems of Linear Equations: The Case of Two Numbers with a Difference and a Sum

Algebra is a powerful tool for solving real-world problems through the use of equations. In this article, we explore the method of solving a specific kind of system of linear equations. We will find the two numbers given that the difference between them is 24 and their sum is 60. This example will introduce you to the step-by-step process of solving such problems.

Introduction to the Problem

The problem at hand requires us to find two numbers where the difference between the larger number (let's call it x) and the smaller number (let's call it y) is 24, and their sum is 60. This can be represented by the following system of equations:

Equation 1: x - y 24

Equation 2: x y 60

Step-by-Step Solution

To solve for the two numbers, we will follow these steps:

**Add the two equations together** to eliminate one of the variables and find the value of the remaining variable.

Equation 1: x - y 24

Equation 2: x y 60

Adding these equations, we get: 2x 84

**Solve for x** by dividing both sides of the equation by 2.

2x 84

x 42

**Substitute the value of x back into one of the original equations** to solve for y.

Using the sum equation (x y 60):

42 y 60

Subtracting 42 from both sides:

y 18

**Verification** - We can verify our solution by checking if it satisfies both original equations:

Equation 1: x - y 24 x 42, y 18 42 - 18 24 (True) Equation 2: x y 60 42 18 60 (True)

Conclusion

Through the step-by-step process of solving this system of linear equations, we discovered that the smaller number is 18 and the larger number is 42. This method can be applied to similar problems where we have the difference and the sum of two numbers and need to find the individual values. Understanding these techniques can greatly enhance one's problem-solving skills in algebra and beyond.