Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of algebra and tackle the equation: $\frac{8u - 2}{7} = \frac{5u - 9}{6} + 4$. Don't worry if it looks a bit intimidating at first; we'll break it down step by step, making it super easy to understand. Solving for u means we need to find the value of the variable u that makes this equation true. Think of it like a puzzle where we have to isolate u on one side of the equation. This is a fundamental skill in mathematics, so let's get started. We'll go through each step, explaining the reasoning behind it, so you not only solve the problem but also understand the process. By the end, you'll be able to confidently solve this type of equation and many others. Ready? Let's go!

Step 1: Eliminate the Fractions

Alright, the first thing we want to do is get rid of those pesky fractions. Fractions can make things look messy, so we'll find a way to clear them out. To do this, we need to find the least common multiple (LCM) of the denominators, which are 7 and 6. The LCM of 7 and 6 is 42. Now, multiply every term in the equation by 42. This is super important because it's what cancels out the denominators. So, the equation becomes:

$42 * \frac{8u - 2}{7} = 42 * \frac{5u - 9}{6} + 42 * 4$

Simplify each term: $(42/7)*(8u-2)$ is $6 * (8u-2)$, $(42/6)*(5u-9)$ is $7 * (5u-9)$, and $42 * 4 = 168$. Thus, the equation becomes

$6 * (8u - 2) = 7 * (5u - 9) + 168$

See how much cleaner that looks? We've successfully removed the fractions, which makes the equation much easier to handle. This step is all about making the equation more manageable by getting rid of the fractions. Always start by identifying the denominators and finding their LCM. Multiplying every term by the LCM is the key to clearing those fractions. This technique is really useful for solving a wide variety of algebraic equations. If you ever have fractions in an equation, the first thing you want to do is clear them out using this method. Got it? Awesome!

Step 2: Expand and Simplify

Now that we've cleared the fractions, the next step is to expand the expressions by multiplying out the terms inside the parentheses. Use the distributive property for this. Remember, this means multiplying the number outside the parentheses by each term inside. Let's do it:

• $6 * 8u = 48u$
• $6 * -2 = -12$
• $7 * 5u = 35u$
• $7 * -9 = -63$

So, our equation now looks like this: $48u - 12 = 35u - 63 + 168$. Now, we want to simplify by combining like terms on the right side of the equation. Combining -63 and 168 gives us 105. Thus, we have the simplified equation as:

$48u - 12 = 35u + 105$

This step is crucial because it simplifies the equation. Expanding the terms ensures that we can combine like terms, making the equation easier to solve. Always remember to distribute correctly, multiplying the outside term by each term inside the parentheses. And when combining like terms, pay attention to the signs – positive and negative signs are important! By carefully expanding and simplifying, we're getting closer to isolating u. Great job, guys!

Step 3: Isolate the Variable Term

Okay, now it's time to get all the terms containing u on one side of the equation and the constants (numbers without u) on the other side. Let's start by subtracting $35u$ from both sides of the equation. This will move the u term from the right side to the left side:

$48u - 35u - 12 = 35u - 35u + 105$

This simplifies to:

$13u - 12 = 105$

Next, to isolate the u term further, we'll add 12 to both sides of the equation. This gets rid of the -12 on the left side:

$13u - 12 + 12 = 105 + 12$

Which simplifies to:

$13u = 117$

This step is all about getting the variable terms together. By performing the same operation (adding or subtracting) on both sides of the equation, we keep the equation balanced. The goal here is to get all the u terms on one side. This is a critical step because it sets up the final step where we'll solve for u. Remember to always perform the same operation on both sides to maintain the equation's balance. You're doing awesome!

Step 4: Solve for u

We're almost there, guys! We have $13u = 117$. To isolate u, we need to get rid of the 13 that's multiplying it. We do this by dividing both sides of the equation by 13:

$\frac{13u}{13} = \frac{117}{13}$

This simplifies to:

$u = 9$

And there you have it! We've found the value of u. The solution to the equation $\frac{8u - 2}{7} = \frac{5u - 9}{6} + 4$ is $u = 9$. To verify our answer, we can substitute $u = 9$ back into the original equation and check if both sides are equal. If they are, then we know our solution is correct. Congratulations! You've successfully solved for u. This is the final step, where we isolate the variable by performing the opposite operation (in this case, division) of what's being done to it. Always remember to perform this operation on both sides of the equation. And don't forget to check your answer by plugging it back into the original equation. That's a great way to make sure your solution is correct. Give yourself a pat on the back – you did it!

Conclusion: Practice Makes Perfect!

So, we've walked through solving the equation $\frac{8u - 2}{7} = \frac{5u - 9}{6} + 4$ step by step. We eliminated fractions, expanded and simplified, isolated the variable term, and finally, solved for u. Solving equations is a fundamental skill in mathematics, and it's something that gets easier with practice. Try solving similar equations on your own. Start with simple ones and gradually increase the difficulty. You'll find that with each equation you solve, you'll become more confident and proficient. Don't be afraid to make mistakes; they're a part of the learning process. The more you practice, the better you'll become. Remember to always double-check your work and to understand the reasoning behind each step. Keep up the great work, and happy solving!

Key Takeaways:

• Clear Fractions First: Always begin by eliminating fractions using the least common multiple (LCM). This simplifies the equation and makes it easier to manage.
• Expand and Simplify: Use the distributive property to expand any parentheses and then combine like terms. This streamlines the equation.
• Isolate the Variable: Move all terms containing the variable to one side and constants to the other. This focuses the equation on the variable you're solving for.
• Solve for the Variable: Isolate the variable by performing the opposite operation. This reveals the solution.
• Check Your Answer: Always substitute your answer back into the original equation to ensure it's correct.

By following these steps, you'll be well on your way to mastering equation-solving. Keep practicing, and you'll become a pro in no time! Good luck, and keep up the great work! Let me know if you have any questions!