Solving Algebraic Equations with Real-World Problems: A Comprehensive Guide

Algebra is a powerful tool that goes beyond abstract mathematics and is applicable to numerous real-world scenarios. This article explores the solution of algebraic problems using the example of two numbers with a specific relationship, and how to find the values of these numbers given their sum. We will also examine the works of George Polya, a renowned mathematician, in solving similar problems and understand the diversity of approaches in mathematical problem-solving.

Solving Algebraic Equations Using Basic Steps

Consider the problem: 'One number is one less than another number. If the sum of the two numbers is 177, what are each of these numbers?' To solve this, we can denote the two numbers as (x) and (y), where (x) is one less than (y).

Step-by-Step Solution

Define the variables:

Let's denote the two numbers as (x) and (y), where (x y - 1).

Set up the equation for the sum of the numbers:

[x y 177]

Substitute the first equation into the second:

[y - 1 y 177]

Simplify and solve for (y):

[2y - 1 177]

[2y 178]

[y 89]

Substitute (y 89) back into the first equation to find (x):

[x y - 1 89 - 1 88]

Thus, the two numbers are (x 88) and (y 89).

George Polya’s Contribution to Problem-Solving

George Polya was a significant figure in the field of mathematics, known for his work in problem-solving and mathematical pedagogy. Here, we present a different interpretation of the same problem using Polya's strategies:

Alternative Solution by George Polya

Given:

[x y - 1]

[x y 177]

Substitute (x y - 1) into the second equation:

[y - 1 y 177]

[2y - 1 177]

Solve for (y):

[2y 178]

[y 89]

Find (x) using (x y - 1):

[x 89 - 1 88]

The two numbers are (x 88) and (y 89).

Multiplying to the Fun: Another Problem Solved

Let's explore another problem: 'One number is 2 less than another number. If the sum of the two numbers is 187, what are each of these numbers?' We will solve it using a similar approach.

Multiplicative Approach

Define the variables:

Let (x) be the larger number, and (y x - 2).

Set up the equation for the sum of the numbers:

[x y 187]

Substitute (y x - 2) into the second equation:

[x (x - 2) 187]

[2x - 2 187]

Solve for (x):

[2x 189]

[x 94.5]

Find (y) using (y x - 2):

[y 94.5 - 2 92.5]

Thus, the two numbers are (x 94.5) and (y 92.5).

Conclusion

George Polya's contributions to problem-solving have profoundly impacted the field of mathematics and education. Whether through algebraic equations or other mathematical techniques, the ability to break down problems into simpler steps and systematically solve them is a valuable skill. Understanding these methods not only enhances one's mathematical proficiency but also fosters critical thinking and logical reasoning.

For additional resources and further exploration of problem-solving techniques, consider exploring more problems in algebraic equations and the works of George Polya.