Solving Equations: A Step-by-Step Guide

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Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of solving equations, specifically the one that's been bugging you: mm−3−4=3m−3\frac{m}{m-3}-4=\frac{3}{m-3}. It might look a little intimidating at first, but trust me, we'll break it down step by step and make it super clear. Our goal here isn't just to find the answer but to really understand how to solve these types of problems. So, buckle up, grab your favorite snacks, and let's get started! We will explore the method and its concepts to fully understand the question and how to solve it. Let's start with the basics.

First, what even is an equation? Well, think of it like a seesaw. Both sides have to be balanced. Whatever you do to one side, you must do to the other to keep it balanced. This fundamental principle is the key to solving equations. In this specific equation, we have fractions, which can sometimes make things a bit tricky. But don't worry; we'll handle them systematically. We'll be using some basic algebraic manipulations, like combining like terms, isolating the variable, and checking for extraneous solutions. An extraneous solution is a value that seems to be a solution when we work through the algebra but doesn't actually work in the original equation. This often happens with equations involving fractions or square roots because the algebra can sometimes introduce values that aren't valid. We'll pay close attention to this as we go along. In the equation mm−3−4=3m−3\frac{m}{m-3}-4=\frac{3}{m-3}, our primary goal is to find the value(s) of 'm' that make the equation true. Let's make sure that you completely understand the concept and method for solving the equation. Remember that the methods you learn here can be used in many other situations, so let's get into it, shall we?

Step-by-Step Solution

Now, let's roll up our sleeves and solve the equation. The equation we're tackling is mm−3−4=3m−3\frac{m}{m-3}-4=\frac{3}{m-3}. Here's a breakdown of how to solve it:

  1. Get rid of the fractions: The easiest way to deal with fractions in an equation is to eliminate them. We can do this by multiplying both sides of the equation by the common denominator. In this case, the common denominator is (m−3)(m-3). So, we multiply both sides by (m−3)(m-3): (m−3)∗(mm−3−4)=(m−3)∗(3m−3)(m-3) * (\frac{m}{m-3}-4) = (m-3) * (\frac{3}{m-3}).

  2. Simplify: Now, let's simplify the equation. On the left side, we distribute (m−3)(m-3) to both terms: m−4(m−3)=3m - 4(m-3) = 3.

  3. Expand and Combine: Expand the equation and combine like terms: m−4m+12=3m - 4m + 12 = 3 −3m+12=3-3m + 12 = 3.

  4. Isolate the Variable: Our next step is to isolate 'm'. Subtract 12 from both sides: −3m=3−12-3m = 3 - 12 −3m=−9-3m = -9.

  5. Solve for m: Finally, divide both sides by -3 to solve for 'm': m=−9/−3m = -9 / -3 m=3m = 3.

So, according to our calculations, m=3m=3. But, hold on a sec, we're not quite done yet! There's one more super important step, which we will see in the next section. Before going forward, let's quickly recap what we did: We started with an equation that included fractions. To get rid of those fractions, we multiplied both sides by the common denominator. This step is super important and simplifies the equation. Then, we simplified the equation, expanded, and combined like terms. Next, we worked to isolate the variable, which in our case is 'm'. This meant getting the 'm' term by itself on one side of the equation. We did this by adding and subtracting numbers from both sides. And finally, we solved for 'm' by dividing both sides by the coefficient of 'm'. This is the basic method, but we have to take care of the last step before being done.

Checking for Extraneous Solutions

Now, here's where things get interesting, guys! Remember how I mentioned extraneous solutions? This is where we need to be extra careful. It's crucial to check our potential solution in the original equation to make sure it's valid. Sometimes, the algebraic steps can introduce solutions that don't actually work in the original equation, especially when we have fractions. So, we need to plug our answer, m=3m=3, back into the original equation, mm−3−4=3m−3\frac{m}{m-3}-4=\frac{3}{m-3}. Let's see what happens:

Substitute m=3m = 3: 33−3−4=33−3\frac{3}{3-3} - 4 = \frac{3}{3-3}.

This simplifies to 30−4=30\frac{3}{0} - 4 = \frac{3}{0}.

Uh oh! Do you see the problem? We have division by zero. Division by zero is undefined in mathematics. This means that m=3m=3 is not a valid solution. When we plugged in our result into the original equation, the denominator became zero, which is not allowed. Because division by zero is undefined, any value of 'm' that makes the denominator equal to zero is not a valid solution. In our original equation, the denominator is (m−3)(m-3). Therefore, if m=3m=3, then the denominator becomes zero, which makes the equation undefined. So, even though we arrived at m=3m=3 through our algebraic manipulations, it's not a valid solution. What does this mean? It means there is no solution to this equation. This is a common situation with equations involving fractions. Always, always check your solutions back in the original equation to make sure they're valid. Checking for extraneous solutions is a super important step. Remember, the goal of solving an equation is to find the values of the variable that make the equation true. In this case, we found that no value of 'm' works, so there is no solution.

Conclusion and Key Takeaways

Alright, folks, we've reached the end of our equation-solving adventure! Let's summarize what we've learned and highlight some key takeaways:

  • Solving Equations is all about keeping the equation balanced. Whatever you do to one side, you must do to the other.
  • Eliminating Fractions: Multiplying both sides by the common denominator is a powerful way to get rid of fractions and simplify the equation.
  • Isolating the Variable: The goal is to get the variable (in our case, 'm') by itself on one side of the equation.
  • Extraneous Solutions: Always, always check your solutions back in the original equation, especially when dealing with fractions or square roots. This ensures that you don't include invalid solutions.
  • No Solution: Sometimes, an equation might have no solution, as we saw in this example. This happens when the algebraic steps lead to a contradiction or an undefined expression (like division by zero).

So, in the equation mm−3−4=3m−3\frac{m}{m-3}-4=\frac{3}{m-3}, we found that there is no solution because our potential solution, m=3m=3, resulted in division by zero. I hope this was helpful! Solving equations is a fundamental skill in math, and with practice, you'll become more and more confident. Keep practicing, keep learning, and don't be afraid to ask for help when you need it. And most importantly, keep that mathematical curiosity alive. If you have any more questions, feel free to ask! Remember the key steps and always check your final answer to see if it makes sense in the original question. Understanding each step thoroughly is essential for solving these types of problems. That's all for today, guys. Have a great day and keep solving equations! Also, don't forget that if the answer is the choice