Solving Algebraic Equations with Real-Life Problems: The Sum of Two Numbers

Introduction to Algebraic Thinking

Algebra is a branch of mathematics that involves solving equations and relating variables to each other. It is a fundamental skill that plays a crucial role in various fields, from engineering to data science. In this article, we will explore how to solve algebraic equations using a real-life problem: the sum of two numbers.

The Problem: Sum of Two Numbers

The problem at hand involves finding two numbers whose sum is 46, and where one number is greater than the other by 4.

Step-by-Step Solution

Let's denote the two numbers as (x) and (y). According to the problem, we have the following two equations:

(x y 46) (x - y 4)

We can solve these equations simultaneously by following these steps:

From the second equation, express (x) in terms of (y):

x  y   4

Substitute this expression for (x) into the first equation:

y   4   y  46

Combine like terms:

2y   4  46

Subtract 4 from both sides:

2y  42

Divide both sides by 2:

y  21

Now that we have (y), we can find (x):

x  y   4  21   4  25

Thus, the two numbers are 25 and 21.

Other Solutions to the Problem

It is important to note that there can be different ways to solve the same problem. Here are a few alternative approaches:

Approach 1

Let one be (x) and the other be (y).

The equations can be written as:

xy 46 xy 4

From the second equation, express (x) in terms of (y):

x  4y

Substitute this expression for (x) into the first equation:

4y y  46

Combine like terms:

5y  46

Solve for (y):

y  46 / 5

Now, find (x):

x  4y  4 * (46 / 5)

However, this approach is more complex and might not yield an integer result, which typically is the case for this problem.

Approach 2

Let the numbers be (y) and (y 4). So:

y   (y   4)  46

Combine like terms:

2y   4  46

Subtract 4 from both sides:

2y  42

Divide both sides by 2:

y  21

From the second equation, we get (x):

x  y   4  21   4  25

Thus, the two numbers are 21 and 25.

Conclusion

While there are multiple ways to solve this problem, the algebraic approach we used provides a clear and systematic method. It emphasizes the importance of variable manipulation and solving simultaneous equations. Understanding these techniques is crucial for solving more complex algebraic problems in various real-life scenarios.

If you are looking to enhance your algebra skills, practice similar problems, or simply want to improve your problem-solving abilities, this article provides a solid foundation. Remember, the key to mastering algebra lies in consistent practice and logical reasoning.