Solving The Equation 3/(m+3) - M/(3-m) = (m^2+9)/(m^2-9) A Step-by-Step Guide

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Introduction: Unraveling the Mathematical Puzzle

In the realm of mathematics, equations serve as intricate puzzles, challenging us to decipher the values that satisfy their conditions. This article delves into the process of solving a specific equation: ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$. We will embark on a step-by-step journey, employing algebraic techniques to isolate the unknown variable, m, and arrive at the solution. This exploration will not only illuminate the solution itself but also enhance our understanding of equation-solving strategies.

Equation solving is a fundamental skill in mathematics, applicable across various disciplines, including physics, engineering, and computer science. The ability to manipulate equations, identify potential solutions, and verify their validity is crucial for problem-solving and decision-making in diverse contexts. In this article, we will not only find the solution to the given equation but also discuss the underlying principles and techniques that empower us to tackle a wide range of mathematical challenges.

Our journey will involve navigating through fractions, factoring expressions, and handling potential extraneous solutions. By the end of this exploration, you will have a solid grasp of how to solve this equation and a broader appreciation for the elegance and power of algebraic methods. So, let's embark on this mathematical adventure and uncover the solution to the equation at hand.

Step 1: Identifying Restrictions and Simplifying the Equation

The first crucial step in solving any equation, especially those involving fractions, is to identify any restrictions on the variable. These restrictions arise from denominators that cannot be equal to zero. In our equation, ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$, we have three denominators: m + 3, 3 - m, and m² - 9. Setting each of these equal to zero, we find the following restrictions:

• m + 3 = 0 => m ≠ -3
• 3 - m = 0 => m ≠ 3
• m² - 9 = 0 => (m + 3)(m - 3) = 0 => m ≠ -3 and m ≠ 3

Therefore, m cannot be equal to 3 or -3. These values would make the denominators zero, resulting in undefined expressions. We must keep these restrictions in mind and check our final solution to ensure it does not violate them.

Next, we simplify the equation by finding a common denominator for all the fractions. Notice that m² - 9 can be factored as (m + 3)(m - 3). Also, 3 - m is the negative of m - 3. We can rewrite the equation as:

${\frac{3}{m+3} + \frac{m}{m-3} = \frac{m^2+9}{(m+3)(m-3)}}$

Now, the least common denominator (LCD) is (m + 3)(m - 3). Multiplying both sides of the equation by the LCD, we eliminate the fractions and obtain a simpler equation to solve.

This step is critical for making the equation more manageable and sets the stage for further algebraic manipulations. By eliminating the fractions, we transform the equation into a more familiar form, allowing us to apply standard techniques for solving polynomial equations. Remember, identifying restrictions and simplifying are essential tools in your equation-solving arsenal.

Step 2: Multiplying by the Common Denominator and Expanding

Having identified the restrictions (m ≠ 3 and m ≠ -3) and found the least common denominator (LCD) as (m + 3)(m - 3), we now proceed to multiply both sides of the equation by the LCD. This step is crucial for clearing the fractions and transforming the equation into a more manageable form. Our equation, ${\frac{3}{m+3} + \frac{m}{m-3} = \frac{m^2+9}{(m+3)(m-3)}}$, becomes:

3(m - 3) + m(m + 3) = m² + 9

This multiplication effectively eliminates the denominators, leaving us with a polynomial equation. The next step is to expand the terms on the left-hand side of the equation. We distribute the 3 and the m across the parentheses:

3m - 9 + m² + 3m = m² + 9

Expanding the terms allows us to combine like terms and simplify the equation further. This process is a fundamental algebraic technique that is used extensively in solving equations and simplifying expressions. The accuracy of this step is paramount, as any errors in expansion will propagate through the rest of the solution.

By carefully multiplying and expanding, we have transformed the equation from a complex fractional form to a simpler polynomial form. This simplification is a significant step towards isolating the variable m and finding the solution. The next stage involves combining like terms and rearranging the equation into a standard form for solving.

Step 3: Combining Like Terms and Simplifying to a Linear Equation

Following the expansion in the previous step, our equation stands as: 3m - 9 + m² + 3m = m² + 9. The next logical step is to combine like terms on the left-hand side of the equation. We have two terms involving m (3m and 3m) and constant terms (-9 and implicitly 0):

Combining the m terms, we get 3m + 3m = 6m. So, the equation becomes:

m² + 6m - 9 = m² + 9

Now, we aim to simplify the equation further by eliminating the m² term from both sides. We can achieve this by subtracting m² from both sides:

m² + 6m - 9 - m² = m² + 9 - m²

This simplifies to:

6m - 9 = 9

Notice that the equation has now been transformed from a quadratic equation (with an m² term) into a linear equation (with only an m term). This simplification is crucial because linear equations are much easier to solve. The goal is to isolate the variable m on one side of the equation.

Simplifying equations by combining like terms and eliminating common terms is a fundamental technique in algebra. It allows us to reduce complex equations to simpler, more manageable forms. This step often involves careful attention to signs and coefficients, ensuring that terms are combined correctly. The transformation to a linear equation in this case makes the subsequent steps much more straightforward.

Step 4: Isolating the Variable and Finding the Potential Solution

Having simplified the equation to a linear form, 6m - 9 = 9, our next objective is to isolate the variable m. This involves performing algebraic operations to get m by itself on one side of the equation. To begin, we add 9 to both sides of the equation:

6m - 9 + 9 = 9 + 9

This simplifies to:

6m = 18

Now, m is being multiplied by 6. To isolate m, we divide both sides of the equation by 6:

${\frac{6m}{6} = \frac{18}{6}}$

This gives us:

m = 3

At this point, we have found a potential solution for m. However, it is crucial to remember the restrictions we identified in Step 1. We found that m cannot be equal to 3 or -3 because these values would make the denominators in the original equation zero, resulting in undefined expressions.

Isolating the variable is the core of solving any equation. It involves using inverse operations to undo the operations that are being performed on the variable. In this case, we used addition and division to isolate m. However, finding a potential solution is not the end of the process. We must always check for extraneous solutions, which are values that satisfy the simplified equation but not the original equation.

Step 5: Checking for Extraneous Solutions and Determining the Final Solution

In the previous step, we arrived at a potential solution for the equation: m = 3. However, as we emphasized earlier, it is imperative to check this solution against the restrictions we identified in Step 1. Recall that the original equation, ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$, had restrictions m ≠ 3 and m ≠ -3. These restrictions are in place because these values would make the denominators of the fractions zero, leading to undefined expressions.

Our potential solution, m = 3, directly violates one of these restrictions. Therefore, m = 3 is an extraneous solution. An extraneous solution is a value that satisfies a transformed equation but not the original equation. This often occurs when we perform operations, such as squaring both sides or multiplying by an expression containing the variable, that can introduce new solutions that are not valid in the original context.

Since m = 3 is an extraneous solution, it means that there is no value of m that satisfies the original equation. The equation has no solution. This outcome highlights the importance of checking for extraneous solutions. Failing to do so can lead to incorrect answers and a misunderstanding of the mathematical problem.

Checking for extraneous solutions is a critical step in solving equations, especially those involving fractions or radicals. It ensures that the solutions we find are valid in the original context of the problem. If a potential solution violates any restrictions or does not satisfy the original equation, it must be discarded. In this case, the extraneous solution led us to the correct conclusion that the equation has no solution.

Conclusion: The Equation's Unsatisfying Solution

In this comprehensive exploration, we embarked on a step-by-step journey to solve the equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$. We meticulously navigated through the process, starting with identifying restrictions on the variable, simplifying the equation by finding a common denominator, expanding terms, combining like terms, isolating the variable, and finally, checking for extraneous solutions.

Our efforts led us to a potential solution of m = 3. However, this value clashed with the initial restrictions we established, as it would result in division by zero in the original equation. Consequently, we identified m = 3 as an extraneous solution, a value that satisfies a transformed equation but not the original one.

The ultimate conclusion is that the equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$ has no solution. There is no value of m that can satisfy the equation without violating the fundamental rules of mathematics.

This exercise underscores the importance of a rigorous approach to equation solving. It is not enough to simply manipulate equations and arrive at a potential answer. We must always be mindful of restrictions and extraneous solutions. These can invalidate our results and lead to incorrect conclusions. By diligently following a systematic process, we can ensure the accuracy and validity of our solutions.

The absence of a solution is a valid outcome in mathematics. It indicates that the conditions imposed by the equation are inherently contradictory or incompatible. In such cases, it is crucial to recognize and acknowledge the lack of a solution, rather than forcing a potentially incorrect answer.

Through this journey, we have not only solved a specific equation but also reinforced essential problem-solving skills and a deeper understanding of algebraic principles. These skills will serve us well in tackling future mathematical challenges, where the ability to think critically and approach problems systematically is paramount.