Solving The Equation 3/(m+3) - M/(3-m) = (m^2+9)/(m^2-9)

Hey guys! Today, we're diving deep into the fascinating world of algebra to tackle a seemingly complex equation. Our mission, should we choose to accept it, is to find the solution to this mathematical puzzle: ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$. Don't worry if it looks intimidating at first glance; we'll break it down step by step, making sure everyone can follow along. So, grab your thinking caps, and let's get started!

Demystifying the Equation

Before we jump into solving, let's take a moment to understand what we're dealing with. The equation ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$ involves fractions with variables in the denominators, which means we need to be extra careful about potential values of ${ m }$ that could make the denominators zero. Remember, dividing by zero is a big no-no in the math world! So, our first step is to identify any restrictions on ${ m }$. We need to ensure that ${ m+3 \neq 0 }$, ${ 3-m \neq 0 }$, and ${ m^2-9 \neq 0 }$. Solving these inequalities will give us the values of ${ m }$ that we must exclude from our final solution set. This initial step is crucial because it sets the stage for a smooth and accurate solving process. Once we've identified these restrictions, we can proceed with manipulating the equation to isolate ${ m }$ and find its possible values. It’s like laying the groundwork before building a house; a solid foundation ensures a sturdy and reliable structure. Understanding the equation also involves recognizing the different parts and how they interact. We have fractions, subtraction, and an equality sign, all working together to create a mathematical statement. Our goal is to unravel this statement and find the value(s) of ${ m }$ that make it true. So, with our restrictions in mind, let's move on to the next phase: simplifying the equation.

Finding the Common Denominator

The heart of solving this equation lies in our ability to combine the fractions. To do that, we need a common denominator. Looking at our equation, ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$, we can see the denominators are ${ m+3 }$, ${ 3-m }$, and ${ m^2-9 }$. The last denominator, ${ m^2-9 }$, should ring a bell; it's a difference of squares! We can factor it as ${ (m+3)(m-3) }$. Now, notice something clever: ${ 3-m }$ is just the negative of ${ m-3 }$. This means we can rewrite the second fraction to have a denominator of ${ m-3 }$ (or ${ m+3 }$, for that matter) and then combine it with the other fractions. The least common denominator (LCD) here is ${ (m+3)(m-3) }$, which is super convenient because it's already present in one of the fractions. Our next move is to rewrite each fraction with this LCD. This involves multiplying the numerator and denominator of the first fraction by ${ (m-3) }$ and the numerator and denominator of the second fraction by ${ -(m+3) }$. This might seem like a bit of algebraic gymnastics, but it's a crucial step in simplifying the equation. By having a common denominator, we can combine the numerators and work towards isolating ${ m }$. It's like merging different streams into a single river; once they flow together, we can navigate more easily. So, let's get those fractions aligned and ready for the next step!

Simplifying and Combining Fractions

Alright, let's roll up our sleeves and dive into the nitty-gritty of simplifying. We've identified our common denominator as ${ (m+3)(m-3) }$, and now it's time to rewrite each fraction with this denominator. Starting with the first fraction, ${\frac{3}{m+3}}$, we multiply both the numerator and denominator by ${ (m-3) }$, giving us ${\frac{3(m-3)}{(m+3)(m-3)}}$. Next up, the second fraction, ${\frac{m}{3-m}}$. Remember that ${ 3-m }$ is the negative of ${ m-3 }$, so we can rewrite this fraction as ${-\frac{m}{m-3}}$. To get the common denominator, we multiply both the numerator and denominator by ${ -(m+3) }$, which turns our fraction into ${\frac{m(m+3)}{(m+3)(m-3)}}$. Now, our equation looks like this: ${\frac{3(m-3)}{(m+3)(m-3)} + \frac{m(m+3)}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. Notice the plus sign? That's because we took care of the negative sign in the original equation. With a common denominator, we can finally combine the fractions on the left side. We add the numerators: ${ 3(m-3) + m(m+3) }$. Expanding this gives us ${ 3m - 9 + m^2 + 3m }$, which simplifies to ${ m^2 + 6m - 9 }$. So, our equation now reads ${\frac{m^2 + 6m - 9}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. We're getting closer! We've successfully combined the fractions and simplified the numerator. The next step is to get rid of those pesky denominators and see what we're left with. Keep your eyes on the prize, guys; we're on the home stretch!

Eliminating the Denominators and Solving

Now for the fun part – eliminating those denominators! We've got our equation in the form ${\frac{m^2 + 6m - 9}{(m+3)(m-3)} = \frac{m^2+9}{(m+3)(m-3)}}$. Since both sides have the same denominator, we can multiply both sides of the equation by ${ (m+3)(m-3) }$. This will cancel out the denominators, leaving us with a much simpler equation: ${ m^2 + 6m - 9 = m^2 + 9 }$. Notice how the fractions have magically disappeared? This is the power of algebraic manipulation! Now, we have a quadratic equation (well, almost!). We can simplify this further by subtracting ${ m^2 }$ from both sides, which gives us ${ 6m - 9 = 9 }$. See? Much more manageable! Our next step is to isolate ${ m }$. We add 9 to both sides of the equation, resulting in ${ 6m = 18 }$. Finally, to solve for ${ m }$, we divide both sides by 6, giving us ${ m = 3 }$. Boom! We've found a potential solution. But hold on a second… remember those restrictions we talked about at the beginning? We need to check if our solution is valid. This is a crucial step that many people overlook, but it can save us from making mistakes. So, before we declare victory, let's check if ${ m = 3 }$ fits the bill.

Checking for Extraneous Solutions

Okay, we've arrived at a potential solution: ${ m = 3 }$. But before we throw a math party, we need to put on our detective hats and check for extraneous solutions. These are sneaky values that satisfy the simplified equation but not the original one, usually because they make a denominator zero. Remember our original equation: ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$? We identified restrictions on ${ m }$ to avoid dividing by zero. Let's revisit those restrictions. We had ${ m+3 \neq 0 }$, ${ 3-m \neq 0 }$, and ${ m^2-9 \neq 0 }$. Solving these inequalities, we find that ${ m \neq -3 }$ and ${ m \neq 3 }$. Uh oh! Our potential solution, ${ m = 3 }$, violates one of these restrictions. This means ${ m = 3 }$ is an extraneous solution. It's a mathematical imposter! So, what does this mean for our equation? It means that there is no value of ${ m }$ that satisfies the original equation. The equation has no solution. This might feel a bit anticlimactic, but it's an important lesson in math: always check your solutions! Extraneous solutions can pop up in equations with fractions or radicals, so it's crucial to be vigilant. We went through all the steps, did the algebra correctly, but our final answer is that there is no solution. Sometimes, that's just the way it is in the world of math. So, we've successfully navigated this equation, identified a potential solution, and then cleverly unmasked it as an extraneous one. Great job, guys! We've conquered this mathematical challenge together.

Conclusion: The Empty Solution Set

Well, guys, we've reached the end of our mathematical journey, and what a journey it has been! We started with a complex-looking equation, ${\frac{3}{m+3}-\frac{m}{3-m}=\frac{m^2+9}{m^2-9}}$, and step by step, we unraveled its mysteries. We found a potential solution, ${ m = 3 }$, but like true math detectives, we didn't stop there. We rigorously checked our solution against the original equation's restrictions and discovered that it was an extraneous solution – a false lead! This means that, despite our best efforts, there is no value of ${ m }$ that satisfies the equation. The solution set is empty, often denoted by the symbol ${ \emptyset }$ or simply stated as "no solution." This outcome highlights an important aspect of problem-solving in mathematics: not every equation has a solution. Sometimes, the conditions imposed by the equation are contradictory, leading to an empty solution set. It's like trying to fit a square peg into a round hole – it just won't work. But don't be discouraged! The process of solving the equation is just as valuable as finding the solution itself. We honed our skills in simplifying fractions, finding common denominators, and checking for extraneous solutions. These are skills that will serve us well in future mathematical endeavors. So, while we didn't find a numerical solution this time, we gained valuable experience and a deeper understanding of algebraic equations. And that, my friends, is a victory in itself! Keep exploring, keep questioning, and keep solving!