Solving Equations: A Step-by-Step Guide

Hey guys! Ever stumble upon an equation that looks like a tangled mess? Don't sweat it! We're diving into the world of equations, specifically tackling the equation: $\frac{1}{7 x}-\frac{1}{x+6}=\frac{1}{3 x^2+18 x}$. This might seem intimidating at first glance, but trust me, with a systematic approach, we can unravel this puzzle together. This guide is all about breaking down complex problems into manageable steps. We'll explore the fundamental concepts and techniques needed to solve equations effectively. We will focus on the given equation and then extend this knowledge to cover other types of equations. So, grab your pencils, and let's get started. We'll start with the basics, build up our skills, and eventually, you'll be solving equations like a pro. This guide is your cheat sheet, your roadmap, and your friend in the world of mathematics. We will walk through this together and make sure that you are up to speed with the fundamentals. The goal here is to make sure you have the tools needed to be successful. Let's make this fun, and learn something new together, okay?

Step 1: Understanding the Problem and Initial Setup

Alright, let's start by looking closely at the equation $\frac{1}{7 x}-\frac{1}{x+6}=\frac{1}{3 x^2+18 x}$. The first thing we should always do is to understand the context of the question. What type of equation are we dealing with? What does the question ask us? In this case, our goal is to find the value (or values) of x that satisfy this equation. The key idea is to isolate x on one side of the equation. Before we jump into the math, it's crucial to identify any potential restrictions on our solution. Here, we must be careful about values of x that would make the denominators equal to zero, as division by zero is undefined. This means x cannot equal 0 or -6. Why? Because these values would result in division by zero in the original equation. We'll keep these restrictions in mind as we work through the steps. Think of this initial check as a safety net – it prevents us from stumbling into nonsensical solutions later on. We must always, always remember to verify the solution we get at the end because we may get answers that violate the original question. A common problem that beginner students have is forgetting this crucial step. We're setting the stage for a solution and must get it right. Before we do anything, let's get our initial conditions down: x ≠ 0 and x ≠ -6. Okay? Now we can start, knowing that we are ready to solve the equation. The initial setup is to rewrite the question, and we're ready to get started. Just like setting up the goal, it is crucial to understand the rules of the game. Alright, let's get this show on the road!

Step 2: Simplifying the Equation

Now, let's simplify the equation to make it easier to solve. The goal here is to get rid of the fractions. To do that, we need to find a common denominator. Look at the denominators in our equation: 7x, (x+6), and 3x^2 + 18x. Notice that 3x^2 + 18x can be factored as 3x(x+6). This is a crucial step! The simplest way to do this is to get the least common multiple or LCM. The LCM of the denominators is 21x(x+6). This means we'll multiply both sides of the equation by 21x(x+6). Here's how it breaks down:

• Multiply each term by 21x(x+6).

  • For the first term, (1/(7x)) * 21x*(x+6) = 3(x+6)
  • For the second term, -1/(x+6) * 21x(x+6) = -21x
  • For the third term, 1/(3x*(x+6)) * 21x(x+6) = 7

This gives us a much cleaner equation: 3(x+6) - 21x = 7. See? Much better!

This step is where the equation begins to transform from something complex into something that we can handle. Remember to take it easy and write down each step carefully to reduce the chance of making mistakes. We are moving closer to our solution, and now we must focus on getting rid of the fractions. We have successfully simplified the equation by identifying the least common multiple or the LCM. Great job! Let's move on and solve our equation now.

Step 3: Solving the Simplified Equation

Awesome, now that we've simplified, let's solve the equation 3(x+6) - 21x = 7. This involves expanding the terms, collecting like terms, and isolating x. Remember, our ultimate goal is to get x by itself on one side of the equation. So let's start by expanding the terms. First, we have 3(x+6), which becomes 3x + 18. This gives us:

3x + 18 - 21x = 7

Next, let's combine the x terms: 3x - 21x = -18x. Now our equation is:

-18x + 18 = 7

To isolate x, we first subtract 18 from both sides: -18x = 7 - 18, which simplifies to -18x = -11.

Finally, divide both sides by -18 to solve for x: x = -11 / -18 = 11/18. Therefore, our solution is x = 11/18.

Well done, we now have a value for x. Remember, the steps here are always the same. Expand the equation first, then collect like terms, and then simplify. It may take some practice at first, but it will get easier with time. We are almost there, just one more step left. Are you ready? Let's check our solution now!

Step 4: Verifying the Solution

Alright, guys, before we celebrate, we must verify our solution! Remember those restrictions we mentioned in Step 1? We need to make sure our solution doesn't violate them. In our case, x = 11/18. Let's revisit our restrictions: x cannot be 0 and x cannot be -6. Since 11/18 is not equal to either of those values, our solution is valid regarding those restrictions. Now, let's plug x = 11/18 back into the original equation to ensure it holds true. This is the most crucial step. If the equation doesn't work with our calculated value for x, we've made an error somewhere. Let's do it: Original equation: $\frac{1}{7 x}-\frac{1}{x+6}=\frac{1}{3 x^2+18 x}$.

Substitute x with 11/18

$\frac{1}{7 *(11/18)}-\frac{1}{(11/18)+6}=\frac{1}{3 (11/18)^2+18 (11/18)}$

Calculating and simplifying both sides, we should get the same value. If the left-hand side equals the right-hand side, then our solution is correct. This step is about accuracy and ensures we are 100% correct in our answer. After carefully computing each term, you will find that the equation is indeed balanced! The left side and right side are equal. This confirms that x = 11/18 is the correct solution. Always double-check your work, and don't skip this important step. In the real world, it's always important to check your results. We're almost done, let's recap the steps.

Step 5: Summary and Conclusion

We did it! We successfully solved the equation $\frac{1}{7 x}-\frac{1}{x+6}=\frac{1}{3 x^2+18 x}$. Let's recap the steps:

1. Understand and Setup: We identified the restrictions and initial setup.
2. Simplify: We simplified the equation by finding a common denominator and eliminating fractions.
3. Solve: We isolated x using algebraic manipulations.
4. Verify: We confirmed our solution by plugging it back into the original equation.

From our initial setup, we had to remember the conditions where the equation is undefined. We learned about the importance of checking our solution. Then, we simplified it to get the solution. Finally, we checked our answer to see whether it was the right solution! And, of course, the answer is x = 11/18.

Solving equations is a fundamental skill in mathematics. The process we used here – understanding the problem, simplifying, solving, and verifying – applies to many other equation types. Keep practicing, and you'll find that solving equations becomes easier and more intuitive. Remember, it's all about breaking down complex problems into manageable steps. Keep practicing, and you'll become a pro in no time! Keep exploring, keep questioning, and keep learning. Math can be fun and challenging, and with the right approach, you can conquer any equation that comes your way! I hope this helps you guys! Let me know if you have any questions!