Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of solving equations, specifically tackling a problem like this: $\frac{x+1}{14}=\frac{4}{7}+\frac{x-8}{6}$. Don't worry if it looks a bit intimidating at first; we'll break it down into manageable steps. The goal is simple: find the value of 'x' that makes the equation true. It's like a puzzle, and we're the detectives! We'll use a combination of algebraic manipulation and a dash of common sense to crack the code. This guide will walk you through each step, making sure you understand the 'why' behind the 'how'. We'll transform this seemingly complex equation into something that's easy to solve. So, grab your pencils, and let's get started. By the end, you'll be able to solve similar equations with confidence. This journey will focus on clarity and practical application, ensuring you not only find the solution but also learn the underlying principles. Ready to become an equation-solving pro? Let's do this!

Step 1: Eliminate the Fractions – The Ultimate Strategy

Our first mission, guys, is to get rid of those pesky fractions. Fractions can be a real pain when you're trying to isolate 'x', but thankfully, there's a simple trick. We'll find the least common multiple (LCM) of the denominators, which are 14, 7, and 6. The LCM is the smallest number that all these numbers divide into evenly.

Let's break down the process. The prime factors of 14 are 2 and 7. The prime factor of 7 is 7. And the prime factors of 6 are 2 and 3. So, to find the LCM, we take the highest power of each prime factor present: 2 (from 14 and 6), 3 (from 6), and 7 (from 14 and 7). Multiplying these together gives us 2 * 3 * 7 = 42. So, the LCM of 14, 7, and 6 is 42. Now, we'll multiply every term in the equation by 42. This is crucial because it clears out the denominators, making our equation much easier to handle. It's like giving the equation a makeover, guys! This step is all about simplifying the equation to make it more manageable. Multiplying each term by the LCM ensures that we maintain the equality of the equation while eliminating the fractions. Remember, what you do to one side, you must do to the other side to keep things balanced. Doing this efficiently is the key to solving this type of equation. Keep it in mind. This is a foundational technique that you'll use over and over again in algebra. Once you get the hang of it, you'll find it incredibly powerful.

Performing the Multiplication

Let's apply this strategy to our equation: $\frac{x+1}{14}=\frac{4}{7}+\frac{x-8}{6}$. Multiplying each term by 42, we get:

• 42 * (x+1)/14 = 42 * (4/7) + 42 * (x-8)/6

Now, let's simplify each term.

• 42/14 simplifies to 3, so the first term becomes 3 * (x+1)
• 42/7 simplifies to 6, so the second term becomes 6 * 4
• 42/6 simplifies to 7, so the third term becomes 7 * (x-8)

This gives us a much cleaner equation: 3(x + 1) = 6 * 4 + 7(x - 8). See how much easier that looks without fractions? By applying this carefully, we've successfully removed the fractions. Now, it's all about simplifying and isolating 'x'. The fractions are gone. Awesome, right? Let's move on to the next step.

Step 2: Expand and Simplify – Time to Unleash the Power of Distribution

Alright, squad, now that we've cleared the fractions, it's time to expand the equation using the distributive property. This means multiplying the numbers outside the parentheses by each term inside the parentheses. Let's do it! Remember, the distributive property states that a(b + c) = ab + ac.

So, let's apply this to our equation: 3(x + 1) = 6 * 4 + 7(x - 8). Expanding the terms, we get:

• 3 * x + 3 * 1 = 24 + 7 * x - 7 * 8
• 3x + 3 = 24 + 7x - 56

See? It's all about multiplication. Now, let's simplify the right side of the equation by combining the constants: 24 - 56 = -32. Therefore, our equation becomes:

• 3x + 3 = 7x - 32

We're making progress. Now we're dealing with a much simpler equation. This step is about getting rid of those parentheses and making the equation easier to solve. Always remember the distributive property. Practice makes perfect. Don't be afraid to double-check your work, especially when multiplying.

Combining Like Terms

Before we move on, let's take a quick look at combining like terms. In our example, we don't have any like terms on each side of the equation to combine, but it's always a good idea to simplify as much as possible at each stage. This makes the next steps easier and reduces the chances of making mistakes. Like terms are terms that have the same variable raised to the same power. For instance, 3x and 7x are like terms, and the numbers 3, 24, and -56 are also like terms. Be careful with those negative signs. Always pay attention to whether you are adding or subtracting. Simplify your terms and you will keep it under control. The better you understand how to combine like terms, the more efficiently you can solve equations.

Step 3: Isolate the Variable – The Grand Finale

Here we are, the final step! Our mission is to isolate 'x' on one side of the equation. This involves moving all the terms with 'x' to one side and all the constant terms (just numbers) to the other side. Let's start by getting all the 'x' terms on the left side. We have 3x on the left and 7x on the right. To move the 7x to the left side, we subtract 7x from both sides. Remember, whatever you do to one side, you must do to the other to keep the equation balanced. This is a fundamental rule in algebra. So, we get:

• 3x - 7x + 3 = 7x - 7x - 32

• -4x + 3 = -32

Now, we need to get rid of the +3 on the left side. To do this, we subtract 3 from both sides:

• -4x + 3 - 3 = -32 - 3

• -4x = -35

Almost there! We've isolated the term with 'x'. Now, all that's left is to solve for 'x'.

Solving for 'x'

To solve for 'x', we need to get 'x' by itself. We have -4x = -35. To isolate 'x', we divide both sides of the equation by -4. Dividing both sides by the same number maintains the equation's balance. This is crucial for arriving at the correct answer. This gives us:

• -4x / -4 = -35 / -4

• x = 35/4 or x = 8.75

And there you have it! We've found the value of 'x'. Awesome job, team! We've successfully solved the equation and found the value of 'x' that makes it true. Always double-check your answer by plugging it back into the original equation to ensure it's correct.

Step 4: Verification - Double-Checking Your Work

Alright, guys, before we celebrate, let's make sure our answer is correct. It's always a smart move to verify your solution. We'll plug our calculated value of x back into the original equation: $\frac{x+1}{14}=\frac{4}{7}+\frac{x-8}{6}$. We found that x = 35/4 or 8.75. Let's substitute x = 8.75 into the equation.

• $\frac{8.75+1}{14}=\frac{4}{7}+\frac{8.75-8}{6}$

• $\frac{9.75}{14}=\frac{4}{7}+\frac{0.75}{6}$

Now, let's simplify and see if both sides are equal.

• 0.6964 = 0.5714 + 0.125

• 0.6964 = 0.6964

Since both sides are equal, our solution is correct. Congratulations! It's always a good idea to verify your solution to catch any potential errors. It's like having a safety net. This step not only confirms your answer but also helps you build confidence in your problem-solving skills. Verification ensures accuracy. If the sides are not equal, then there is a mistake that should be checked in the previous steps.

Conclusion

And there you have it! We've successfully navigated the world of equation solving. You've learned how to clear fractions, expand and simplify, isolate the variable, and verify your answer. Solving equations is a fundamental skill in mathematics, and now you have the tools to tackle similar problems with confidence. Keep practicing, and you'll become a pro in no time! Remember, the key is to break down the problem into smaller, more manageable steps. Each step is a building block toward the solution. Don't be afraid to take your time and double-check your work. Practice makes perfect. Keep up the great work and keep exploring the amazing world of mathematics! You've got this! Keep practicing different types of equations. You will become even better at solving equations. Embrace the challenge. Remember to review the steps we covered to solve the equations. This understanding will boost your confidence in problem-solving. Stay curious, keep learning, and enjoy the journey!