Solving The Algebraic Equation 1/(x-2) + 1/(x^2-7x+10) = 6/(x-2) A Step-by-Step Guide
Introduction: Navigating the Realm of Algebraic Equations
In the captivating world of mathematics, algebraic equations stand as fundamental pillars, serving as gateways to problem-solving and analytical thinking. This article delves into the intricacies of solving a specific algebraic equation: 1/(x-2) + 1/(x^2-7x+10) = 6/(x-2). We embark on a journey to unravel the solution, employing a blend of algebraic manipulation, factorization techniques, and meticulous analysis. Our aim is not merely to arrive at the solution but to provide a comprehensive understanding of the underlying principles and strategies involved.
At the heart of our exploration lies the given equation: 1/(x-2) + 1/(x^2-7x+10) = 6/(x-2). This equation, characterized by its rational expressions, presents a unique challenge that demands careful consideration. To effectively tackle this equation, we must navigate the terrain of fractions, factoring, and the crucial identification of potential extraneous solutions. Each step will be meticulously explained, ensuring that readers of all mathematical backgrounds can grasp the concepts and techniques employed.
Before diving into the solution, it is paramount to acknowledge the significance of understanding the domain of the equation. The denominators of the fractions, (x-2) and (x^2-7x+10), play a pivotal role in defining the permissible values of x. We must tread cautiously to avoid values that render the denominators zero, as such values would result in undefined expressions. Therefore, we will first determine the domain of the equation, laying the groundwork for a valid solution.
Step 1: Defining the Domain – A Foundation for a Valid Solution
The domain of an equation encompasses the set of all permissible values for the variable x. In the context of our equation, the domain is constrained by the denominators of the fractions. We must exclude values of x that would make the denominators zero, as division by zero is an undefined operation. Let's analyze the denominators:
- Denominator 1: (x-2). This denominator becomes zero when x = 2. Therefore, x = 2 is excluded from the domain.
- Denominator 2: (x^2-7x+10). To find the values that make this denominator zero, we factor the quadratic expression: (x^2-7x+10) = (x-2)(x-5). This denominator becomes zero when x = 2 or x = 5. Consequently, both x = 2 and x = 5 are excluded from the domain.
Therefore, the domain of the equation is all real numbers except x = 2 and x = 5. This crucial understanding will guide us in verifying the validity of our solutions later on.
Step 2: Simplifying the Equation – A Quest for Common Ground
To effectively manipulate the equation, we seek a common denominator for the fractions. This allows us to combine the terms and simplify the expression. Observing the denominators, (x-2) and (x^2-7x+10), we recognize that (x^2-7x+10) can be factored as (x-2)(x-5). This revelation unveils the common denominator: (x-2)(x-5).
Now, we rewrite the equation with the common denominator:
[1/(x-2)] * [(x-5)/(x-5)] + 1/[(x-2)(x-5)] = [6/(x-2)] * [(x-5)/(x-5)]
This transformation yields:
(x-5)/[(x-2)(x-5)] + 1/[(x-2)(x-5)] = 6(x-5)/[(x-2)(x-5)]
With the common denominator in place, we can combine the numerators:
(x-5 + 1)/[(x-2)(x-5)] = 6(x-5)/[(x-2)(x-5)]
Simplifying the numerator on the left side, we get:
(x-4)/[(x-2)(x-5)] = 6(x-5)/[(x-2)(x-5)]
Step 3: Eliminating the Denominator – A Step Towards a Simpler Equation
With the denominators now identical, we can eliminate them by multiplying both sides of the equation by the common denominator, (x-2)(x-5). This step effectively clears the fractions and transforms the equation into a more manageable form:
[(x-4)/[(x-2)(x-5)]] * [(x-2)(x-5)] = [6(x-5)/[(x-2)(x-5)]] * [(x-2)(x-5)]
This simplification leads to:
x-4 = 6(x-5)
Step 4: Solving the Linear Equation – Unveiling the Solution
We now have a linear equation, which is significantly easier to solve. Expanding the right side, we get:
x-4 = 6x - 30
To isolate x, we subtract x from both sides:
-4 = 5x - 30
Next, we add 30 to both sides:
26 = 5x
Finally, we divide both sides by 5:
x = 26/5
Step 5: Verifying the Solution – A Crucial Step for Accuracy
Before declaring our solution, we must verify its validity by checking if it falls within the domain of the equation. Recall that the domain excludes x = 2 and x = 5. Our solution, x = 26/5, which is 5.2, is not equal to either of these excluded values. Therefore, it is a valid solution.
Additionally, we can substitute x = 26/5 back into the original equation to confirm that it satisfies the equation. This step provides further assurance of the accuracy of our solution.
Conclusion: The Triumph of Algebraic Techniques
In this article, we have successfully navigated the intricacies of solving the equation 1/(x-2) + 1/(x^2-7x+10) = 6/(x-2). Through a combination of algebraic manipulation, factorization, and careful consideration of the domain, we arrived at the solution x = 26/5. This journey underscores the power of algebraic techniques in unraveling mathematical challenges. The steps involved, from defining the domain to verifying the solution, highlight the importance of a meticulous approach in problem-solving.
Key takeaways from this exploration include:
- The significance of understanding the domain of an equation to ensure the validity of solutions.
- The power of factorization in simplifying complex expressions.
- The importance of verifying solutions to prevent extraneous results.
By mastering these concepts and techniques, you can confidently tackle a wide range of algebraic equations and unlock the beauty and power of mathematics.
This comprehensive guide aims to empower readers with the knowledge and skills necessary to solve similar equations and foster a deeper appreciation for the world of mathematics. Remember, the journey of mathematical exploration is a continuous one, and each problem solved is a step forward in expanding our understanding and capabilities.
