Solving \(\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}\) A Step-by-Step Guide

Introduction

In this article, we will delve into the intricate steps required to solve the equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$. This equation, which combines rational expressions, requires a meticulous approach to ensure accuracy. Understanding how to solve rational equations like this is fundamental in algebra, as it involves various algebraic techniques such as finding common denominators, simplifying expressions, and solving quadratic equations. We will break down each step, providing a clear and concise explanation to aid understanding and mastery. This guide is designed for students, educators, and anyone looking to enhance their problem-solving skills in mathematics. Mastering the art of solving such equations not only improves mathematical proficiency but also enhances analytical thinking, a skill invaluable in various fields. In the subsequent sections, we will explore each step in detail, from identifying the restrictions on the variable ${ m }$ to verifying the solutions obtained. Our goal is to transform a seemingly complex problem into a series of manageable steps, making it accessible to all learners. By the end of this guide, you should be well-equipped to tackle similar challenges and confidently apply these techniques to solve a wide range of algebraic equations. The journey of solving equations like this is not just about finding the answer; it's about developing a deep understanding of the underlying mathematical principles.

1. Identifying Restrictions on ${ m }$

Before diving into the algebraic manipulation, it is crucial to identify any restrictions on the variable ${ m }$. Restrictions arise from denominators in the equation, as division by zero is undefined. In our equation, ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$, we have three denominators to consider: ${ m+3 }$, ${ 3-m }$, and ${ m^2-9 }$. Setting each of these equal to zero will help us determine the values of ${ m }$ that are not permissible. First, we examine ${ m+3 }$. Setting this equal to zero gives us ${ m+3 = 0 }$, which implies ${ m = -3 }$. Thus, ${ m }$ cannot be ${ -3 }$. Next, we consider ${ 3-m }$. Setting this equal to zero, we have ${ 3-m = 0 }$, which gives us ${ m = 3 }$. Therefore, ${ m }$ cannot be ${ 3 }$. Finally, we analyze ${ m^2-9 }$. Setting this equal to zero, we get ${ m^2-9 = 0 }$. This is a difference of squares, which can be factored as ${ (m+3)(m-3) = 0 }$. This equation is satisfied when either ${ m+3 = 0 }$ or ${ m-3 = 0 }$. Solving these gives us ${ m = -3 }$ and ${ m = 3 }$, respectively. These values reiterate our earlier findings. In summary, the restrictions on ${ m }$ are ${ m \neq -3 }$ and ${ m \neq 3 }$. These restrictions are critical because if we obtain these values as solutions later in the process, we must discard them. By identifying these restrictions early on, we ensure that our final solutions are valid and do not lead to undefined expressions in the original equation. This step highlights the importance of careful preliminary analysis in solving rational equations. Neglecting these restrictions can lead to incorrect solutions, making this a vital part of the problem-solving process. The identification of restrictions underscores a key principle in algebra: the need to be mindful of the domain of variables and the potential for undefined expressions.

2. Finding the Common Denominator

The next critical step in solving the equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$ is to find a common denominator for all the fractions. This will allow us to combine the terms and simplify the equation. The common denominator is essential for adding and subtracting fractions, just as it is with numerical fractions. To find the common denominator, we first need to factor all the denominators. We already know that ${ m+3 }$ and ${ 3-m }$ are linear terms and thus cannot be factored further. However, ${ m^2-9 }$ is a difference of squares and can be factored as ${ (m+3)(m-3) }$. Now we have the three denominators: ${ m+3 }$, ${ 3-m }$, and ${ (m+3)(m-3) }$. To determine the common denominator, we need to include each factor the greatest number of times it appears in any one denominator. The factors are ${ (m+3) }$ and ${ (m-3) }$. Notice that ${ 3-m }$ is the negative of ${ m-3 }$, which means we can rewrite the term ${\frac{m}{3-m}}$ as ${-\frac{m}{m-3}}$. This will simplify the process of finding the common denominator. Therefore, the common denominator is ${ (m+3)(m-3) }$. With the common denominator identified, we can now rewrite each fraction with this denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factors to achieve the common denominator. The goal is to transform the equation into a form where we can easily combine the fractions and proceed with solving for ${ m }$. The careful selection of the common denominator is pivotal in simplifying the equation and making it easier to solve. This step is not just a mechanical process; it requires an understanding of algebraic principles and the properties of fractions. A clear grasp of this concept is crucial for handling more complex equations involving rational expressions. This process of finding a common denominator sets the stage for further algebraic manipulations and ultimately solving the equation. The next step involves rewriting each fraction with the common denominator and combining the terms, which we will discuss in the subsequent section.

3. Rewriting Fractions with the Common Denominator

With the common denominator established as ${ (m+3)(m-3) }$, the next step is to rewrite each fraction in the original equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$ with this denominator. Rewriting fractions with a common denominator is a fundamental technique in algebra that allows us to combine and simplify expressions. First, let's rewrite the equation, addressing the sign issue in the second term as discussed earlier: ${\frac{3}{m+3} + \frac{m}{m-3} = \frac{m^2+9}{(m+3)(m-3)}}$. Now, we need to adjust the numerators of the first two fractions so that they have the common denominator ${ (m+3)(m-3) }$. For the first fraction, ${\frac{3}{m+3}}$, we multiply both the numerator and the denominator by ${ (m-3) }$: ${\frac{3(m-3)}{(m+3)(m-3)} = \frac{3m-9}{(m+3)(m-3)}}$. For the second fraction, ${\frac{m}{m-3}}$, we multiply both the numerator and the denominator by ${ (m+3) }$: ${\frac{m(m+3)}{(m-3)(m+3)} = \frac{m^2+3m}{(m+3)(m-3)}}$. The third fraction, ${\frac{m^2+9}{(m+3)(m-3)}}$, already has the common denominator, so we don't need to change it. Now our equation looks like this: ${\frac{3m-9}{(m+3)(m-3)} + \frac{m^2+3m}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. By rewriting each fraction with the common denominator, we have set the stage for combining the numerators. This is a crucial step because it transforms the equation into a form where we can eliminate the denominators and work with a simpler algebraic expression. The process of rewriting fractions requires careful attention to detail to ensure that we are multiplying both the numerator and the denominator by the correct factors. This skill is essential not only for solving equations but also for simplifying complex algebraic expressions in various mathematical contexts. In the next section, we will combine the numerators and simplify the equation further, moving closer to finding the solution for ${ m }$. This step-by-step approach ensures clarity and accuracy in solving the equation.

4. Combining Numerators and Simplifying

Having rewritten each fraction with the common denominator ${ (m+3)(m-3) }$, we can now combine the numerators in the equation ${\frac{3m-9}{(m+3)(m-3)} + \frac{m^2+3m}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. Combining numerators is a key step in simplifying rational equations, allowing us to consolidate terms and reduce the complexity of the equation. Since all the fractions now have the same denominator, we can add the numerators on the left side of the equation: ${\frac{(3m-9) + (m^2+3m)}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. Combine like terms in the numerator on the left side: ${\frac{m^2 + 6m - 9}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. Now that we have a single fraction on each side of the equation with the same denominator, we can eliminate the denominators. This is a valid step as long as the denominator is not zero, which we have already addressed by identifying the restrictions on ${ m }$ earlier. We multiply both sides of the equation by ${ (m+3)(m-3) }$: ${ m^2 + 6m - 9 = m^2 + 9 }$. This step simplifies the equation significantly, transforming it from a rational equation into a simpler polynomial equation. The simplification process involves careful attention to detail to ensure that all terms are correctly combined and that no algebraic errors are made. This is a crucial skill in algebra, applicable to a wide range of problems. By combining the numerators and eliminating the denominators, we have reduced the equation to a more manageable form. In the next step, we will solve this simplified equation for ${ m }$. The ability to simplify equations is a cornerstone of mathematical problem-solving, and this step demonstrates the power of algebraic manipulation in making complex problems more accessible. The journey from the original rational equation to this simplified form highlights the importance of each step in the process.

5. Solving the Simplified Equation

After simplifying the equation, we now have ${ m^2 + 6m - 9 = m^2 + 9 }$. Solving this simplified equation is the next crucial step in finding the solution to the original problem. This step involves using algebraic techniques to isolate the variable ${ m }$ and determine its value(s). First, we observe that there is an ${ m^2 }$ term on both sides of the equation. We can subtract ${ m^2 }$ from both sides to eliminate this term: ${ m^2 + 6m - 9 - m^2 = m^2 + 9 - m^2 }$, which simplifies to ${ 6m - 9 = 9 }$. Now, we have a linear equation in terms of ${ m }$. To isolate ${ m }$, we first add 9 to both sides of the equation: ${ 6m - 9 + 9 = 9 + 9 }$, which simplifies to ${ 6m = 18 }$. Next, we divide both sides by 6 to solve for ${ m }$: ${ \frac{6m}{6} = \frac{18}{6} }$, which gives us ${ m = 3 }$. At this point, it is crucial to remember the restrictions we identified at the beginning of the problem. We found that ${ m }$ cannot be equal to ${ 3 }$ or ${ -3 }$ because these values would make the denominators of the original equation equal to zero, resulting in undefined expressions. Since our solution is ${ m = 3 }$, and this value is a restriction, we must discard it. This means that the original equation has no solution. The process of solving the simplified equation highlights the importance of algebraic manipulation and the careful application of mathematical principles. However, it also underscores the critical need to check our solutions against any restrictions on the variable. Discarding extraneous solutions is a vital part of solving rational equations, ensuring that our final answer is valid. In this case, the absence of a solution underscores that not all equations have solutions, and it's essential to consider the domain of the variables involved. The final step, checking against restrictions, brings the problem-solving process full circle, reinforcing the importance of a thorough and meticulous approach.

6. Verifying the Solution and Final Answer

Having obtained a potential solution of ${ m = 3 }$, and recalling the restrictions ${ m \neq -3 }$ and ${ m \neq 3 }$, verifying the solution is an essential final step. In this case, our potential solution coincides with one of the restrictions, making it an extraneous solution. This means that there is no value of ${ m }$ that satisfies the original equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$. To formally conclude, we state that the equation has no solution. The importance of this verification step cannot be overstated. In solving rational equations, it is common to encounter extraneous solutions, which are values that satisfy the simplified equation but not the original equation due to the restrictions on the variable. Failing to check for extraneous solutions can lead to incorrect conclusions. In this specific problem, the algebraic manipulations were correctly executed, but the solution obtained was invalidated by the initial restrictions. This highlights the interconnectedness of the steps in the problem-solving process. The final answer, “no solution,” is a valid and important conclusion. It indicates that the equation, as presented, does not have any values of ${ m }$ that make it true. This understanding is crucial in mathematics, where recognizing the absence of a solution is as important as finding one. The entire process, from identifying restrictions to solving the equation and verifying the solution, exemplifies a comprehensive approach to problem-solving in algebra. It demonstrates the need for careful attention to detail, a thorough understanding of algebraic principles, and the importance of checking one's work. This methodical approach is not only valuable in mathematics but also in various other problem-solving contexts. The journey through this equation serves as a valuable lesson in the nuances of algebraic problem-solving.

Conclusion

In summary, we have thoroughly explored the process of solving the equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$. Solving rational equations involves a series of steps, each requiring careful attention to detail. We began by identifying the restrictions on the variable ${ m }$, recognizing that the denominators cannot be zero. This led us to the critical restrictions of ${ m \neq -3 }$ and ${ m \neq 3 }$. Next, we found the common denominator, which allowed us to rewrite the fractions and combine them. This simplification process transformed the equation into a more manageable form. After combining the numerators and eliminating the denominators, we arrived at a simplified equation that we could solve for ${ m }$. The solution we obtained, ${ m = 3 }$, coincided with one of the restrictions we had identified earlier. This necessitated a crucial step: verifying the solution against the restrictions. Since our solution was a restricted value, we concluded that the original equation has no solution. This outcome underscores the importance of always checking for extraneous solutions in rational equations. The entire process highlights several key concepts in algebra, including the manipulation of fractions, the identification of restrictions, and the verification of solutions. It also illustrates the importance of a methodical approach to problem-solving, where each step builds upon the previous one to arrive at a logical conclusion. The absence of a solution is itself a valid answer, and recognizing this is a vital aspect of mathematical understanding. The skills and techniques applied in this problem are broadly applicable to other algebraic challenges, making this a valuable exercise in mathematical problem-solving. The journey through this equation serves as a testament to the power of algebraic reasoning and the importance of thoroughness in mathematical analysis.