Let's dive into solving this equation step by step. Equations like this are fundamental in algebra, and mastering them is crucial for tackling more complex math problems. So, grab your pencils, and let's get started, guys! Understanding how to manipulate equations and isolate variables is a key skill that you’ll use throughout your mathematical journey. We'll break down each step, explaining the logic behind it, so you not only get the answer but also understand why we do what we do. By the end of this article, you'll be able to approach similar problems with confidence and ease. Remember, practice makes perfect, so don't hesitate to try out more examples on your own! The initial equation we're tackling is 3 - 2v + 7 = -6v - 14. This might look a bit intimidating at first, but don't worry; we'll simplify it piece by piece. Our goal is to find the value of 'v' that makes this equation true. To do that, we need to isolate 'v' on one side of the equation. This involves several steps, including combining like terms and using inverse operations to move terms around. First, we’ll focus on simplifying each side of the equation separately before we start moving things across the equals sign. This will make the process cleaner and less prone to errors. So, let's jump right into the first step: combining the constants on the left side of the equation.

Step 1: Combine Like Terms

Okay, first things first, let's simplify both sides of the equation. On the left side, we have constants 3 and 7. When combined, they give us 10. So, the left side of the equation becomes 10 - 2v. Now, our equation looks like this: 10 - 2v = -6v - 14. See? Already a bit simpler! Combining like terms is a fundamental step in solving equations. It helps to reduce the number of terms and makes the equation easier to work with. In this case, we combined the constants (the numbers without any variables) on the left side. This is a straightforward addition, but it's important to pay attention to the signs. For example, if we had 3 - 7, we would end up with -4. So, always double-check your arithmetic to avoid mistakes. Now that we've simplified the left side, let's take a look at the right side. In this particular equation, the right side, -6v - 14, is already in its simplest form since -6v and -14 are not like terms. Remember, like terms are terms that have the same variable raised to the same power. Since -6v has the variable 'v' and -14 is a constant, we can't combine them. This means we can move on to the next step, which involves isolating the variable 'v'. We'll do this by moving all the terms with 'v' to one side of the equation and all the constants to the other side. This is where the magic of inverse operations comes in handy!

Step 2: Isolate the Variable

Alright, now let's get those 'v' terms together. We can add 6v to both sides of the equation to move the -6v from the right side to the left. This gives us: 10 - 2v + 6v = -6v - 14 + 6v. Simplifying this, we get 10 + 4v = -14. Remember, the goal here is to get all the terms with 'v' on one side of the equation. To do this, we use the concept of inverse operations. Adding 6v to both sides is the inverse operation of subtracting 6v. When we add 6v to -6v, they cancel each other out, leaving us with just the constant term on the right side. It's crucial to perform the same operation on both sides of the equation to maintain the balance. Think of an equation like a scale; if you add something to one side, you must add the same thing to the other side to keep it balanced. Now that we have 10 + 4v = -14, we need to isolate the 'v' term further. This means we need to get rid of the 10 on the left side. To do this, we subtract 10 from both sides of the equation: 10 + 4v - 10 = -14 - 10. This simplifies to 4v = -24. We're getting closer! By subtracting 10 from both sides, we've successfully isolated the term with 'v' on the left side. Now, all that's left to do is to solve for 'v' by dividing both sides by the coefficient of 'v', which is 4. So, let's move on to the final step and find the value of 'v'.

Step 3: Solve for v

Okay, we're in the home stretch! We have 4v = -24. To solve for 'v', we need to divide both sides of the equation by 4. This gives us: 4v / 4 = -24 / 4. Simplifying this, we find that v = -6. And there you have it! We've solved for 'v'. The value of 'v' that makes the original equation true is -6. To recap, we started with the equation 3 - 2v + 7 = -6v - 14. We combined like terms, isolated the variable, and finally solved for 'v'. Each step involved using inverse operations to maintain the balance of the equation. Remember, the key to solving equations is to stay organized and pay attention to the signs. Double-check your work as you go along to avoid making mistakes. Now that we've found the value of 'v', it's a good idea to check our answer by plugging it back into the original equation. This will ensure that our solution is correct. So, let's do that now. Substitute v = -6 into the original equation: 3 - 2(-6) + 7 = -6(-6) - 14. Simplifying this, we get: 3 + 12 + 7 = 36 - 14. Which further simplifies to: 22 = 22. Since both sides of the equation are equal, we can be confident that our solution, v = -6, is correct. Great job, guys! You've successfully solved the equation. Keep practicing, and you'll become a pro at solving all sorts of algebraic equations. This whole process reinforces not just how to get the answer, but also the importance of checking your work to ensure accuracy. Understanding these steps is essential for building a strong foundation in algebra.

Conclusion

So, to wrap things up, we found that v = -6 is the solution to the equation 3 - 2v + 7 = -6v - 14. Remember, the key steps were combining like terms, isolating the variable by using inverse operations, and then solving for the variable. Always double-check your work to make sure your solution is correct. This exercise is a great example of how to approach linear equations, which are fundamental in mathematics and many real-world applications. The skills you've practiced here – combining like terms, using inverse operations, and checking your solutions – are transferable to more complex problems. Think of solving equations as a puzzle; each step brings you closer to the solution. And just like any puzzle, the more you practice, the better you become at solving it. So, don't be discouraged if you find it challenging at first. Keep practicing, and you'll see improvement over time. And remember, math isn't just about finding the right answer; it's about understanding the process and developing problem-solving skills that you can apply in many different areas of life. Whether you're balancing your budget, calculating the tip at a restaurant, or designing a building, math is all around us. So, embrace the challenge, keep learning, and have fun with it! And don't forget, if you ever get stuck, there are plenty of resources available online and in libraries to help you out. Keep up the great work, guys! You're doing awesome! Always remember that persistence and a positive attitude are key to success in mathematics and in life. So, keep practicing, keep learning, and never give up on your goals. You got this! Solving equations is a fundamental skill, and with practice, it will become second nature. Understanding the 'why' behind each step is just as important as getting the correct answer. Keep exploring and challenging yourself with new problems to solidify your understanding and build confidence.