Solving a System of Equations: A Comprehensive Guide for SEO

This article provides a detailed approach to solving the given system of equations and is designed to be optimized for search engines (SEO). By including relevant keywords such as 'equations', 'algebraic solutions', and 'Google SEO', this content aims to improve its discoverability and relevance on search engines.

Introduction

In this article, we explore the solution to a specific system of equations and provide a step-by-step guide to solving equations of this form. The content is optimized for search engines to enhance its visibility and authority on the topic of algebraic equations.

The System of Equations

We start with the system of equations provided:

Let a ≥ b ≥ c. For this system, we have the following equations:

For a ≥ 0: 69a^3 - a^2 b (Equation 1) 69b^3 - b^2 c (Equation 2) 69c^3 - c^2 0 (Equation 3)

Additionally, we consider special cases such as when a 0 and b c.

Solution Approach

Case 1: When a 0

First, we examine the case when a 0 and substitute it into the equations:

69a^3 - a^2 b → 0 - 0 b → b 0 (Equation 4) 69b^3 - b^2 c → 69(0)^3 - (0)^2 c → c 0 (Equation 5)

From Equations 4 and 5, we see that when a 0, we have:

a 0, b 0, c 0

Case 2: When a ≥ c

Next, we consider the scenario where a ≥ c and subtract Equation 2 from Equation 1:

69b^3 - b^2 c (Equation 2)

69a^3 - a^2 b (Equation 1)

Subtracting Equation 2 from Equation 1:

69a^3 - 69b^3 - a^2 b^2 b - c (Equation 6)

Rearranging and simplifying:

69(b^3 - a^3) (b^2 - a^2) b - c (Equation 7)

Factoring the differences of cubes and squares:

69(b - a)(b^2 ab a^2) (b - a)(b a) b - c (Equation 8)

Since a ≥ b ≥ c, the terms (b - a) and (b^2 ab a^2) and (b a) are non-negative:

69(b - a)(b^2 ab a^2) (b - a)(b a) 0 (Equation 9)

This leads to a contradiction unless:

(b - a)(69(b^2 ab a^2) (b a)) 0 (Equation 10)

Since the second term in the product is always positive, the only solution is:

b - a 0 → b a (Equation 11)

Given b ≥ c, this implies:

a b c (Equation 12)

Case 3: Solving for a

When we have a ≥ c and b a, we substitute b a into the equations:

69a^3 - a^2 - a 0 (Equation 13)

This is a cubic equation in terms of a and can be simplified further:

69a^3 - a^2 - a 0 (Equation 14)

Rearranging:

a(69a^2 - a - 1) 0 (Equation 15)

This gives us two possible solutions:

a 0 69a^2 - a - 1 0

Solving the quadratic equation 69a^2 - a - 1 0 using the quadratic formula:

a -frac{-1}{2 times 69} ± sqrt{left(frac{-1}{2 times 69}right)^2 frac{1}{69}}

a frac{1}{138} ± sqrt{frac{1}{18492} frac{1}{69}}

a frac{1}{138} ± sqrt{frac{1 269}{18492}}

a frac{1}{138} ± sqrt{frac{270}{18492}}

a frac{1}{138} ± sqrt{frac{135}{9246}}

a frac{1}{138} ± sqrt{frac{5}{322}}

a frac{1}{138} ± frac{sqrt{5}}{sqrt{322}}

a frac{1}{138} frac{sqrt{5}}{sqrt{322}} text{ or } frac{1}{138} - frac{sqrt{5}}{sqrt{322}}

a frac{1 sqrt{5 times 322}}{138} text{ or } frac{1 - sqrt{5 times 322}}{138}

a frac{1 sqrt{1610}}{138} text{ or } frac{1 - sqrt{1610}}{138}

Given a ≥ 0, the only valid solution is:

a frac{1 sqrt{1610}}{138}

Conclusion

Through detailed analysis and step-by-step solving, we have determined that the valid solutions to the given system of equations are:

a 0 a frac{1 sqrt{1610}}{138}

Our solutions provide a comprehensive understanding of the system and can be verified by substituting back into the original equations. This approach is crucial for SEO optimization, as it ensures the content is well-structured and optimized for search engines.

Keywords and SEO Optimization

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equations algebraic solutions Google SEO

By including these keywords, the content can be more effectively indexed and retrieved by search engines, leading to improved visibility and relevance on search results pages.