Solving Complex Mathematical Systems Using Algebra and Polynomial Equations

Solving mathematical systems, particularly those involving polynomial equations, can be both intricate and rewarding. In this article, we explore a detailed step-by-step method to solve a specific algebraic system using algebraic manipulation and polynomial equations. This article is targeted at students, teachers, and individuals with an interest in advanced mathematics.

Introduction to the System

The system under discussion is:

xy x^2 / y^2 7

(x - y) * (x^2) / y^2 12

We start by introducing a substitution to simplify the equations: Let w x^2 / y^2.

Step-by-Step Solution

Step 1: Substitution and Simplification

From the first equation:

xy 7 - x^2 / y^2 7 - w

From the second equation:

x - y 12 / w

By combining these two results, we get:

2x 7 - w - 12 / w

2y 7 - w - 12 / w

Solving for w:

w (7w - w^2 - 12^2) / (2w)

This simplifies to a fifth-degree polynomial equation:

w^5 - 15w^4 - 87w^3 - 193w^2 - 24w - 144 0

Using an online graphing calculator, such as WolframAlpha, we find one real solution: w ≈ 6.171713628765.

Using this value of w, we determine the corresponding x and y values:

x ≈ 1.38632

y ≈ -0.55803

Checking the solution:

xy ≈ 1.38632 * -0.55803 ≈ 7.00009

(x - y) * (x^2) / y^2 ≈ 12.00014

These values satisfy the original system of equations, confirming our solution.

Alternative Case Analysis

Case 1: w 3

Let w x^2 / y^2 3 3 implies x ±sqrt{3}y.

Substituting into the system:

x - y 3 7

±sqrt{3}y - y 3 7

y(-1 ± sqrt{3}) 4

y 4 / (-1 ± sqrt{3}) (4 / (-1 ± sqrt{3})) * (1 ± sqrt{3}) / (1 ± sqrt{3})

y (4(1 ± sqrt{3})) / 2 2 ± 2sqrt{3}

x ±sqrt{3}y 6 ± 2sqrt{3}

Case 2: w 4

Let w x^2 / y^2 4 4 implies x ±2y.

Substituting into the system:

x - y 4 7

±2y - y 4 7

y(-1 ± 2) 3

y 3 / (-1 ± 2) 3 or -1

x ±2y 6 or 2

The solutions are:

x, y ∈ {2, -1, 6, 3, 6 - 2sqrt{3}, 2 - 2sqrt{3}, 6 2sqrt{3}, 2 2sqrt{3}}

Conclusion

Understanding and solving complex mathematical systems involves careful algebraic manipulation and solving polynomial equations. In this case, we successfully addressed a challenging system of equations through detailed substitution and polynomial solving techniques. These methods can be applied to similar problems in advanced mathematics and are valuable for students and professionals alike.