Math Problem: Solve For Summation Variables A Through G

Hey math enthusiasts! Let's dive into a fun problem where we'll be solving for some variables to find the sum. We're going to work through an equation step-by-step, and it's all about simplifying and understanding how to manipulate algebraic expressions. This isn't just about getting the right answer; it's about seeing how the pieces fit together. So, buckle up, grab your pens, and let's get started. We'll break down the equation, identify each part, and explain the thought process behind each step. It's like a puzzle, and we're the detectives, figuring out the solution piece by piece. Don't worry if it seems a bit tricky at first; we'll go through it slowly. The goal is to build your confidence and make you feel comfortable with algebra. By the end, you'll be able to tackle similar problems with ease. Let's make math fun and interesting. This is a journey, and we're in this together. Ready? Let's go!

Understanding the Basics: The Setup

Alright, guys, before we jump into the thick of it, let's take a look at the equation. We are given an equation that involves fractions with variables. The whole point is to manipulate these fractions and combine them into a single fraction. We'll be using some basic algebraic rules, like factoring and combining like terms. Our main goal is to simplify the equation and find the sum. Remember, we are looking for the sum of the fractions, and we have to identify the correct values for a, b, and c to solve the equation. The equation is set up to guide us through a series of steps, and each step will bring us closer to the solution. This is a common strategy in algebra, where you are given a problem to solve using step-by-step instructions. Let's start with the equation provided: rac{3 x}{2 x-6}+rac{9}{6-2 x}=rac{3 x}{2 x-6}+rac{9}{a(2 x-6)}. The first thing you'll notice is that we need to find the value of ‘a’. We must manipulate the second fraction in such a way that it has the same denominator as the first fraction. This involves factoring out a -1 from the denominator of the second fraction. This is the foundation we will build upon, so understanding this is critical.

Now, let's look at the given equation rac{3 x}{2 x-6}+rac{9}{6-2 x}=rac{3 x}{2 x-6}+rac{b}{2 x-6}. Here, we will work with the second term on the left side of the equation. We already figured out that the second term must have the same denominator as the first term, which is (2x - 6). However, the original second term has a denominator of (6 - 2x). We need to transform the fraction so that the denominator is the same as the first fraction. This requires a small but important manipulation. The whole idea is to get the denominators to match so that we can easily add the fractions together. Once the denominators are the same, we can then combine the numerators. Keep an eye on how we will adjust the numerator to match our changes in the denominator. This is a key part of solving this kind of problem and will make the next steps much easier to grasp.

Finally, we have rac{3 x}{2 x-6}+rac{b}{2 x-6}=rac{3 x-c}{2 x-6}. In this last step, we are going to work on the numerator and simplify the entire equation. So we have to combine the fractions because they share the same denominator. Once the fractions have a common denominator, you simply combine the numerators over that common denominator. The value of $c$ will become clear once you properly add the terms in the numerator. The goal here is to arrive at a simplified single fraction. This step brings us to the final answer. Remember, the ultimate aim is to make the equation simple and easy to understand. We are now heading towards the end of our math journey; let's figure out the values for a, b, and c and find the sum!

Step-by-Step Solution: Finding the Variables

Okay, guys, let's get down to business and solve for our variables. We'll go through each step carefully and see how everything fits together. The key here is to follow the instructions step by step. This approach ensures that we don’t miss anything important and keeps us organized. Remember, in math, it's often the small details that make the difference. Let's start with the first part of the equation: rac{3 x}{2 x-6}+rac{9}{6-2 x}=rac{3 x}{2 x-6}+rac{9}{a(2 x-6)}. The objective is to make the denominator of the second fraction match the denominator of the first fraction. To do that, we have to change the sign of the terms in the denominator of the second fraction. The given fraction is rac{9}{6-2 x}, and we want to change this into something with the form $a(2x-6)$. To achieve this, we can factor out a -1 from the denominator of the second fraction: $6 - 2x$ becomes $-1(2x - 6)$. Therefore, $a$ should be -1. So, we've got the first piece of the puzzle! Remember, we factored out a -1. This is a crucial step.

Next, let’s consider the equation: rac{3 x}{2 x-6}+rac{9}{6-2 x}=rac{3 x}{2 x-6}+rac{b}{2 x-6}. Because we have already determined that a equals -1, the denominator $6 - 2x$ can now be expressed as $-1(2x-6)$. So, the fraction is now rac{9}{-1(2 x-6)}. We now know that the equation now can be expressed as rac{3x}{2x-6} + rac{9}{-1(2x-6)} = rac{3x}{2x-6} + rac{b}{2x-6}. This means that the numerator 9 must be divided by -1, and b will become -9. Therefore, b is -9. Good job, guys!

Now, let's move on to the final step where we have to simplify. We are given the equation rac{3 x}{2 x-6}+rac{b}{2 x-6}=rac{3 x-c}{2 x-6}. Since the fractions have the same denominator, we can combine the numerators. In the previous step, we found that b is -9. Thus, the equation becomes rac{3 x - 9}{2x-6} = rac{3x - c}{2x-6}. Now, we can see that the numerators are also equivalent. Therefore, $3x - 9 = 3x - c$. This means that $c = 9$. We have now solved for all the variables!

Calculating the Sum and Final Thoughts

Alright, guys, we've found all our values for a, b, and c! We've done the hard work, and now it's time to put it all together. So, to recap, we have $a = -1$, $b = -9$, and $c = 9$. We've taken an equation and broken it down into manageable parts. The main idea here is not just to get the answer, but also to understand the 'why' behind each step. Now, let’s calculate the sum of the variables: $a + b + c = -1 + (-9) + 9$. This gives us a final answer of -1. We can see that by carefully working through each step, we've managed to find the sum. Understanding the concepts of factoring, combining fractions, and manipulating equations will help you in future math problems. The ability to break down a complex problem into smaller parts and solve it systematically is an important skill in math and in life. Feel confident in your new algebra skills. Keep practicing, keep exploring, and keep the curiosity alive.

We started with a set of fractions and ended up simplifying it. We successfully determined the values for the variables a, b, and c, leading us to our final sum of -1. Congratulations, everyone! Remember, the key is to practice, ask questions, and never be afraid to make mistakes. Keep up the great work, and you'll become math masters in no time! Keep practicing, and don’t be afraid to take on challenges. You've got this! Now, go forth and conquer more math problems!