Solving The Equation 3/(a-8) + 2/(a-3) = 0 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. It's a process that involves finding the values of variables that satisfy a given equation. Today, we will delve into solving a specific type of equation: $\frac{3}{a-8}+\frac{2}{a-3}=0$. This equation involves fractions and variables in the denominator, which adds a layer of complexity. However, with a systematic approach, we can break it down and find the solution. This guide will provide a step-by-step solution, detailed explanations, and key concepts to ensure a comprehensive understanding of the process. The main key to solve these equations is understanding how to handle fractions, how to find a common denominator, and how to solve resulting algebraic equations. We'll walk through each step carefully, making sure that you grasp the logic and reasoning behind every operation. By the end of this article, you'll not only know how to solve this particular equation but also gain a solid foundation for tackling similar problems. So, let's embark on this mathematical journey and unlock the solution together.
Understanding the Equation
Before we dive into the solution, let's first understand the structure of the equation $\frac{3}{a-8}+\frac{2}{a-3}=0$. This equation is a rational equation, which means it involves algebraic fractions. The variable 'a' appears in the denominators of these fractions. This is a crucial point because it introduces restrictions on the possible values of 'a'. Specifically, the denominators cannot be zero, as division by zero is undefined in mathematics. Therefore, we must consider the values of 'a' that would make the denominators zero. In this case, a cannot be 8 (because 8 - 8 = 0) and a cannot be 3 (because 3 - 3 = 0). These values are called excluded values or restrictions. They are essential to identify before solving the equation, as any solution we find must not coincide with these excluded values. Ignoring these restrictions can lead to incorrect solutions or misunderstandings of the equation's behavior. The equation's structure also tells us that we will need to combine these fractions. To do so, we'll need to find a common denominator. This will allow us to add the fractions together and simplify the equation. Understanding this preliminary step is vital for efficiently solving the equation. Recognizing the type of equation and its constraints sets the stage for a successful problem-solving approach. By being mindful of these details, we ensure our solution is mathematically sound and logically consistent.
Step-by-Step Solution
Now, let's proceed with the step-by-step solution to the equation $\frac{3}{a-8}+\frac{2}{a-3}=0$.
Step 1: Find a Common Denominator
To combine the two fractions, we need a common denominator. The common denominator for (a - 8) and (a - 3) is their product, which is (a - 8)(a - 3). We multiply the first fraction by $\frac{a-3}{a-3}$ and the second fraction by $\frac{a-8}{a-8}$:
This gives us:
Step 2: Combine the Fractions
Now that the fractions have a common denominator, we can combine them:
Step 3: Simplify the Numerator
Expand and simplify the numerator:
Step 4: Set the Numerator to Zero
For a fraction to be zero, the numerator must be zero (and the denominator must not be zero). So, we set the numerator equal to zero:
Step 5: Solve for 'a'
Solve the linear equation for 'a':
Step 6: Check for Excluded Values
Recall that 'a' cannot be 8 or 3. Our solution, a = 5, is not an excluded value. Therefore, it is a valid solution.
Verifying the Solution
To ensure our solution is correct, we can substitute a = 5 back into the original equation: $\frac{3}{a-8}+\frac{2}{a-3}=0$. Substituting a = 5, we get:
Since the equation holds true, our solution a = 5 is correct. Verifying the solution is a crucial step in solving equations. It confirms that the value we found indeed satisfies the original equation. This process involves plugging the solution back into the equation and ensuring that both sides of the equation are equal. This step is particularly important in rational equations, like the one we solved, because we have to be mindful of excluded values. By verifying, we guard against potential errors and gain confidence in our solution. This practice reinforces the understanding of the equation's properties and the solution's validity. It's a habit that strengthens problem-solving skills and ensures accuracy in mathematical computations. Thus, verification is an indispensable part of the equation-solving process, adding a layer of certainty to the final answer.
Common Mistakes to Avoid
When solving equations, especially rational equations like $\frac{3}{a-8}+\frac{2}{a-3}=0$, it's essential to be aware of common mistakes. One frequent error is neglecting to identify excluded values. Remember, any value of 'a' that makes the denominator zero is an excluded value and cannot be a solution. Forgetting this can lead to incorrect answers. Another common mistake is incorrectly combining fractions. It's crucial to find a common denominator before adding or subtracting fractions. Skipping this step or doing it incorrectly can lead to significant errors in the solution. Additionally, be cautious when simplifying expressions. Ensure you distribute terms correctly and combine like terms accurately. A small mistake in simplification can throw off the entire solution. It's also important to verify the solution by substituting it back into the original equation. This helps catch any algebraic errors made during the solving process. Furthermore, be mindful of the signs when dealing with negative numbers. A sign error can easily occur and change the outcome. Lastly, avoid the temptation to cross-multiply directly without first combining the fractions. Cross-multiplication is a valid technique, but it should be applied after the equation has been properly set up with a single fraction on each side. Being mindful of these common pitfalls can improve accuracy and proficiency in solving rational equations. By developing a systematic approach and paying close attention to detail, you can minimize the chances of making these mistakes.
Alternative Methods
While we've solved the equation $\frac{3}{a-8}+\frac{2}{a-3}=0$ using a standard method, there are alternative approaches that can be employed. One such method involves cross-multiplication. After combining the fractions into a single fraction, we have $\frac{5a - 25}{(a-8)(a-3)} = 0$. Instead of just setting the numerator to zero, we can recognize that a fraction is zero if and only if its numerator is zero (provided the denominator is not zero). Thus, we focus on solving $5a - 25 = 0$, which is a more direct path to the solution a = 5. Another approach is to isolate one of the fractions on one side of the equation and then cross-multiply. This can be particularly useful when dealing with simpler rational equations. For instance, we could rewrite the original equation as $\frac{3}{a-8} = -\frac{2}{a-3}$. Cross-multiplying then gives us $3(a-3) = -2(a-8)$, which simplifies to $3a - 9 = -2a + 16$. Solving this linear equation also leads to the solution a = 5. Additionally, for those familiar with graphical methods, one could graph the function $y = \frac{3}{a-8}+\frac{2}{a-3}$ and find the x-intercepts, which represent the solutions to the equation. This visual approach can provide a different perspective and help confirm the algebraic solution. Each of these methods offers a slightly different way to tackle the problem. Understanding multiple approaches can enhance problem-solving flexibility and provide a deeper understanding of the underlying mathematical principles.
Conclusion
In conclusion, solving the equation $\frac{3}{a-8}+\frac{2}{a-3}=0$ involves a systematic approach that includes finding a common denominator, combining fractions, simplifying the numerator, and solving for the variable. We've seen that the solution is a = 5, which we verified by substituting it back into the original equation. This process highlights the importance of understanding the structure of rational equations and being mindful of excluded values. Common mistakes, such as neglecting excluded values or incorrectly combining fractions, can lead to errors. Therefore, a careful and methodical approach is crucial. We also explored alternative methods, such as cross-multiplication and graphical techniques, which offer different perspectives and can enhance problem-solving skills. Mastering the techniques for solving rational equations is a valuable skill in mathematics. It not only helps in solving specific problems but also builds a foundation for more advanced mathematical concepts. The ability to manipulate algebraic fractions and solve equations is essential in various fields, including physics, engineering, and economics. By understanding the underlying principles and practicing regularly, you can become proficient in solving these types of equations. Remember, mathematics is a journey of continuous learning and practice. Each problem solved adds to your understanding and strengthens your skills. With dedication and perseverance, you can conquer even the most challenging equations. This exploration of solving rational equations serves as a testament to the power and beauty of mathematics in unraveling complex problems and finding elegant solutions.
