Solving M/(m+4) + 4/(4-m) = M^2/(m^2-16) A Step-by-Step Guide

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Rational equations, which involve fractions with variables in the denominator, can seem daunting at first. However, with a systematic approach, these equations can be solved effectively. This article delves into the step-by-step process of solving the rational equation m/(m+4) + 4/(4-m) = m2/(m2-16), providing a clear understanding of the underlying concepts and techniques. Understanding solving rational equations is crucial in various fields of mathematics and its applications, ranging from algebra and calculus to physics and engineering. This guide aims to equip you with the necessary knowledge and skills to confidently tackle similar problems.

The key to solving any rational equation lies in eliminating the fractions. This is achieved by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions involved. The LCD is the smallest expression that is divisible by all the denominators. Once the fractions are cleared, the equation transforms into a simpler form, typically a linear or quadratic equation, which can be solved using standard algebraic techniques. However, it is imperative to check for extraneous solutions after finding the solutions. Extraneous solutions are those that satisfy the transformed equation but not the original rational equation. These solutions usually arise when a solution makes one of the denominators in the original equation equal to zero, rendering the expression undefined. Therefore, the final step in solving a rational equation is always to verify the solutions by substituting them back into the original equation.

Before diving into the specifics of solving our example equation, let's briefly review the fundamental concepts and steps involved in solving rational equations in general. The first step is always to identify the denominators in the equation. Next, find the LCD of these denominators. This often involves factoring the denominators to identify common factors. Once the LCD is found, multiply both sides of the equation by it. This will eliminate the fractions, resulting in a simpler equation. Solve the resulting equation using appropriate algebraic techniques. This may involve solving a linear equation, a quadratic equation, or a higher-degree polynomial equation. Finally, and most importantly, check your solutions by substituting them back into the original equation. Discard any solutions that make any of the denominators equal to zero. With a solid grasp of these steps, you'll be well-prepared to solve a wide range of rational equations.

1. Identifying the Equation

Our specific equation to be solved is:

mm+4+44−m=m2m2−16\frac{m}{m+4} + \frac{4}{4-m} = \frac{m^2}{m^2-16}

This equation is a rational equation because it involves algebraic fractions where the variable 'm' appears in the denominators. To effectively solve this equation, we will follow a series of steps, ensuring we address any potential pitfalls along the way. The first crucial step in solving any rational equation is identifying the restrictions on the variable. Restrictions are values of the variable that would make any of the denominators equal to zero. These values must be excluded from the solution set, as division by zero is undefined. In our equation, the denominators are (m+4), (4-m), and (m^2-16). Setting each of these equal to zero and solving for 'm' will give us the restrictions. This preliminary step is vital to ensure we don't include extraneous solutions in our final answer. Ignoring these restrictions can lead to incorrect solutions and a misunderstanding of the equation's behavior.

Understanding the restrictions is not just about avoiding division by zero; it also provides valuable insights into the nature of the equation. For instance, it can help us identify vertical asymptotes if we were to graph the corresponding rational function. Furthermore, being mindful of the restrictions helps in interpreting the solutions in the context of a real-world problem, where certain values might not be physically meaningful. For example, if 'm' represents the length of a side of a triangle, it cannot be negative or zero. Therefore, the process of identifying restrictions is an integral part of solving rational equations and understanding their implications. By meticulously determining these restrictions, we set the stage for a more accurate and meaningful solution.

In our given equation, the restrictions become apparent when we analyze the denominators. The denominator (m+4) becomes zero when m = -4. The denominator (4-m) becomes zero when m = 4. The denominator (m^2-16) is a difference of squares, which can be factored as (m+4)(m-4). This denominator becomes zero when m = -4 or m = 4. Thus, we have two critical restrictions: m cannot be equal to -4 or 4. These values must be excluded from our solution set. By clearly identifying these restrictions at the outset, we avoid the pitfall of accepting extraneous solutions later on. This proactive approach ensures the integrity of our solution and our understanding of the equation's constraints.

2. Finding the Least Common Denominator (LCD)

The next pivotal step in solving the rational equation is to determine the Least Common Denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators in the equation. Identifying the LCD is crucial because it allows us to clear the fractions, transforming the equation into a simpler, more manageable form. To find the LCD, we first need to factor all the denominators completely. Factoring enables us to identify common factors and construct the LCD by including each factor the greatest number of times it appears in any one denominator. In our equation, the denominators are (m+4), (4-m), and (m^2-16). As we discussed earlier, (m^2-16) can be factored as (m+4)(m-4). The other denominators, (m+4) and (4-m), are already in their simplest forms. However, it's important to note that (4-m) can be rewritten as -(m-4). This slight manipulation will be helpful when we construct the LCD.

The process of finding the LCD is analogous to finding the least common multiple (LCM) of integers. Just as the LCM is the smallest integer divisible by a set of integers, the LCD is the smallest expression divisible by a set of algebraic expressions. The key is to include each unique factor present in the denominators and to use the highest power of each factor that appears in any one denominator. For instance, if the denominators were (x+1), (x+1)^2, and (x-2), the LCD would be (x+1)^2(x-2). Understanding this principle is fundamental to correctly determining the LCD and successfully clearing fractions in rational equations. A carefully constructed LCD is the cornerstone of simplifying rational equations and paving the way for an accurate solution.

In our specific case, the denominators are (m+4), (4-m), and (m^2-16), which factors into (m+4)(m-4). To construct the LCD, we need to include each unique factor: (m+4) and (m-4). Since (m+4) appears once in the first denominator and once in the factored third denominator, we include it once in the LCD. Similarly, (m-4) appears in the factored third denominator and can be obtained from (4-m) by factoring out a -1. Therefore, we include (m-4) once in the LCD. Thus, the LCD for our equation is (m+4)(m-4). This LCD will be instrumental in clearing the fractions and simplifying the equation for further steps. Choosing the correct LCD is crucial; a wrong LCD will complicate the process and may lead to incorrect solutions.

3. Multiplying by the LCD and Simplifying

Once we have determined the LCD, which in our case is (m+4)(m-4), the next step is to multiply both sides of the equation by this LCD. This crucial step eliminates the fractions, transforming the rational equation into a more manageable algebraic equation. Multiplying each term of the equation by the LCD essentially "cancels out" the denominators, leaving us with an equation that is free of fractions. However, it is vital to distribute the LCD carefully and ensure that each term is multiplied correctly. This step often involves simplification, such as canceling common factors between the LCD and the denominators. A meticulous approach at this stage is key to avoiding errors and ensuring the accuracy of the subsequent steps.

The process of multiplying by the LCD is a direct application of the multiplicative property of equality, which states that multiplying both sides of an equation by the same non-zero quantity preserves the equality. This property is the foundation for solving many types of equations, including rational equations. By multiplying by the LCD, we are essentially multiplying both sides of the equation by an expression that contains the denominators of the fractions. This cleverly chosen multiplier allows us to eliminate the fractions in a systematic way. The resulting equation, now free of fractions, is typically a polynomial equation that can be solved using familiar algebraic techniques. Mastering this step is essential for solving rational equations efficiently and accurately.

In our equation, we have:

mm+4+44−m=m2m2−16\frac{m}{m+4} + \frac{4}{4-m} = \frac{m^2}{m^2-16}

Multiplying both sides by the LCD, (m+4)(m-4), we get:

(m+4)(m−4)∗[mm+4+44−m]=(m+4)(m−4)∗m2(m+4)(m−4)(m+4)(m-4) * [\frac{m}{m+4} + \frac{4}{4-m}] = (m+4)(m-4) * \frac{m^2}{(m+4)(m-4)}

Distributing the LCD on the left side, we have:

(m+4)(m−4)∗mm+4+(m+4)(m−4)∗44−m=(m+4)(m−4)∗m2(m+4)(m−4)(m+4)(m-4) * \frac{m}{m+4} + (m+4)(m-4) * \frac{4}{4-m} = (m+4)(m-4) * \frac{m^2}{(m+4)(m-4)}

Now, we simplify by canceling common factors. Note that (4-m) = -(m-4):

m(m−4)−4(m+4)=m2m(m-4) - 4(m+4) = m^2

This simplifies to:

m2−4m−4m−16=m2m^2 - 4m - 4m - 16 = m^2

Combining like terms, we get:

m2−8m−16=m2m^2 - 8m - 16 = m^2

This resulting equation is a quadratic equation, albeit a simplified one. The fractions have been successfully eliminated, and we are now ready to solve for 'm'. The accuracy of this step is paramount, as any errors here will propagate through the rest of the solution. Therefore, careful distribution and simplification are essential for obtaining the correct answer.

4. Solving the Resulting Equation

After multiplying by the LCD and simplifying, we have arrived at the equation:

m2−8m−16=m2m^2 - 8m - 16 = m^2

This is a quadratic equation, but it can be simplified further. The 'm^2' terms on both sides of the equation cancel each other out, which simplifies the equation significantly. This simplification is a common occurrence in rational equations, and it often leads to a linear equation, which is much easier to solve. However, it is important to be aware that sometimes the resulting equation will remain a quadratic or even a higher-degree polynomial equation, requiring the use of appropriate techniques such as factoring, the quadratic formula, or other methods for solving polynomial equations. The key is to carefully analyze the resulting equation after clearing the fractions and choose the most efficient method for solving it.

The process of solving the resulting equation involves isolating the variable 'm' on one side of the equation. This is achieved by performing algebraic operations such as addition, subtraction, multiplication, and division on both sides of the equation. The goal is to manipulate the equation until 'm' is by itself on one side, giving us the solution. In the case of a linear equation, this typically involves a few straightforward steps. For quadratic equations, we might need to factor, complete the square, or use the quadratic formula. The specific method used will depend on the form of the equation. Regardless of the method, the underlying principle is to maintain the equality by performing the same operation on both sides.

In our specific equation, m2−8m−16=m2m^2 - 8m - 16 = m^2, we can subtract 'm^2' from both sides:

m2−8m−16−m2=m2−m2m^2 - 8m - 16 - m^2 = m^2 - m^2

This simplifies to:

−8m−16=0-8m - 16 = 0

Now we have a linear equation. To solve for 'm', we first add 16 to both sides:

−8m−16+16=0+16-8m - 16 + 16 = 0 + 16

This gives us:

−8m=16-8m = 16

Finally, we divide both sides by -8:

−8m−8=16−8\frac{-8m}{-8} = \frac{16}{-8}

So, we find:

m=−2m = -2

This value of m = -2 is a potential solution to our original rational equation. However, as we emphasized earlier, it is crucial to check this solution against the restrictions we identified in the first step. The next step will involve verifying whether m = -2 is indeed a valid solution or an extraneous one.

5. Checking for Extraneous Solutions

The final and crucial step in solving rational equations is to check for extraneous solutions. As previously discussed, extraneous solutions are those that satisfy the transformed equation (after clearing fractions) but do not satisfy the original rational equation. These solutions typically arise because multiplying by the LCD can introduce solutions that make the denominators in the original equation equal to zero, which is undefined. Therefore, it is absolutely essential to verify each solution obtained by substituting it back into the original equation. This step ensures that our solutions are valid and that we haven't inadvertently included any values that would make the original equation undefined. The process of checking for extraneous solutions is a vital safeguard against incorrect answers.

The concept of extraneous solutions highlights a fundamental difference between solving algebraic equations and solving rational equations. In algebraic equations, any solution obtained through valid algebraic manipulations is a genuine solution. However, in rational equations, the presence of variables in the denominators introduces the possibility of extraneous solutions. This is because the domain of a rational expression is restricted to values that do not make the denominator equal to zero. Multiplying by the LCD eliminates the denominators, but it also effectively expands the domain of the equation. This expansion can lead to solutions that are valid in the transformed equation but not in the original equation. Therefore, the check for extraneous solutions is not merely a formality; it is an integral part of the solution process for rational equations.

In our case, we found a potential solution of m = -2. The original equation is:

mm+4+44−m=m2m2−16\frac{m}{m+4} + \frac{4}{4-m} = \frac{m^2}{m^2-16}

We need to substitute m = -2 into this equation and see if it holds true.

Substituting m = -2, we get:

−2−2+4+44−(−2)=(−2)2(−2)2−16\frac{-2}{-2+4} + \frac{4}{4-(-2)} = \frac{(-2)^2}{(-2)^2-16}

Simplifying each term:

−22+46=44−16\frac{-2}{2} + \frac{4}{6} = \frac{4}{4-16}

−1+23=4−12-1 + \frac{2}{3} = \frac{4}{-12}

−1+23=−13-1 + \frac{2}{3} = -\frac{1}{3}

To verify, we can convert -1 to a fraction with a denominator of 3:

−33+23=−13\frac{-3}{3} + \frac{2}{3} = -\frac{1}{3}

−13=−13-\frac{1}{3} = -\frac{1}{3}

Since the equation holds true when m = -2, and -2 is not among our restricted values (-4 and 4), we can conclude that m = -2 is a valid solution.

Conclusion

In conclusion, the solution to the rational equation mm+4+44−m=m2m2−16\frac{m}{m+4} + \frac{4}{4-m} = \frac{m^2}{m^2-16} is m = -2. We arrived at this solution by systematically following a series of steps: identifying the restrictions on the variable, finding the least common denominator (LCD), multiplying both sides of the equation by the LCD to clear the fractions, solving the resulting equation, and crucially, checking for extraneous solutions. This comprehensive approach ensures the accuracy and validity of the solution. Solving rational equations requires careful attention to detail, particularly when handling denominators and checking for extraneous solutions. By understanding the underlying principles and following a structured process, these equations can be solved confidently and accurately.

The process we've outlined is applicable to a wide range of rational equations. The key is to break down the problem into manageable steps and to meticulously execute each step. Remember to always identify restrictions on the variable before beginning the solution process, as this will help you identify and discard any extraneous solutions. Finding the LCD correctly is essential for clearing fractions and simplifying the equation. Solving the resulting equation may involve techniques for linear, quadratic, or higher-degree polynomial equations. And finally, always check your solutions by substituting them back into the original equation. This final step is not just a formality; it is a critical safeguard against errors.

By mastering the techniques presented in this article, you'll be well-equipped to tackle a variety of rational equations. The ability to solve rational equations is a valuable skill in mathematics and its applications. It demonstrates a deep understanding of algebraic principles and problem-solving strategies. With practice and a systematic approach, solving rational equations can become a straightforward and rewarding mathematical exercise. Remember to focus on understanding the underlying concepts, rather than just memorizing steps, and you'll be well on your way to mastering this important mathematical skill.