Solve 10 - 6v = -104: A Step-by-Step Guide
Hey guys! Let's dive into the world of linear equations, specifically focusing on how to solve equations where you have a variable multiplied by a number and then added to or subtracted from another number. It might sound complicated, but trust me, it's super manageable once you break it down. We're going to dissect the equation 10 - 6v = -104 piece by piece, so you'll be a pro at solving these in no time! Think of this equation as a puzzle, and our goal is to figure out what value of v makes the equation true. We'll use some basic algebraic principles to isolate v on one side of the equation. So, grab your pencils, and let's get started on this mathematical adventure!
Understanding the Equation: 10 - 6v = -104
Before we jump into solving, let's really understand what this equation is telling us. At its heart, this equation is a statement that two things are equal. On the left side, we have 10 - 6v, which means we're starting with 10 and subtracting 6 times some unknown value v. The right side, -104, is just a number. Our mission, should we choose to accept it (and we do!), is to find the exact value of v that makes the left side equal to -104. The -6v part means that the variable v is being multiplied by -6. The 10 is a constant term, just hanging out on its own. It's super important to recognize these individual components because they dictate the steps we'll take to solve the equation. We need to isolate the term with v first, and then we can isolate v itself. This understanding of the structure of the equation is key to successfully navigating the solution. Remember, algebra is like learning a new language, and understanding the vocabulary (terms, coefficients, constants) makes everything else flow more smoothly.
Step 1: Isolating the Variable Term
The first step in solving this equation, 10 - 6v = -104, is to isolate the term that contains our variable, which is -6v. To do this, we need to get rid of the 10 that's being added to the -6v. The golden rule of algebra is that whatever you do to one side of the equation, you must do to the other. This keeps the equation balanced, like a perfectly balanced scale. Since we have +10 on the left side, we're going to do the opposite operation, which is subtraction. We'll subtract 10 from both sides of the equation. So, we have:
10 - 6v - 10 = -104 - 10
This simplifies to:
-6v = -114
See how the 10 and -10 on the left side canceled each other out? That's exactly what we wanted! Now we have the term with the variable (-6v) all by itself on the left side. We're one step closer to unlocking the value of v. The key here is to always perform the opposite operation to move terms around. If something is being added, subtract it. If something is being subtracted, add it. This principle is the foundation of solving algebraic equations, and mastering it will make you a true equation-solving superstar. Don't be afraid to take it slow and double-check your work to ensure you're keeping that equation perfectly balanced.
Step 2: Solving for the Variable
Now that we've isolated the variable term, -6v = -114, the next step is to solve for v itself. Remember, -6v means -6 multiplied by v. To undo this multiplication and isolate v, we need to perform the opposite operation, which is division. We're going to divide both sides of the equation by the coefficient of v, which is -6. This is where paying attention to signs becomes super important! So, we divide both sides by -6:
-6v / -6 = -114 / -6
On the left side, -6 divided by -6 is just 1, so we're left with 1v, which is the same as v. On the right side, we have -114 divided by -6. A negative number divided by a negative number results in a positive number. When we do the division, we find that -114 / -6 = 19. Therefore, our equation simplifies to:
v = 19
We've done it! We've successfully solved for v. This means that the value of v that makes the original equation true is 19. It's like we've cracked the code and found the missing piece of the puzzle. Always remember that division is the key to undoing multiplication, and vice versa. Understanding these inverse operations is crucial for solving equations efficiently and accurately. Plus, you get that awesome feeling of accomplishment when you finally find the solution!
Verification: Checking Your Solution
Before we celebrate our mathematical victory too much, it's always a good idea to verify our solution. This means plugging the value we found for v (which is 19) back into the original equation to make sure it actually works. This step is like a final exam for our solution – it confirms whether we got it right or if we need to go back and check our work. Our original equation was:
10 - 6v = -104
Now we substitute v with 19:
10 - 6(19) = -104
Let's simplify the left side. First, we do the multiplication: 6 * 19 = 114. So we have:
10 - 114 = -104
Now, subtract 114 from 10:
-104 = -104
Look at that! The left side equals the right side. This confirms that our solution, v = 19, is indeed correct. We aced the test! Verification is a crucial step in problem-solving, not just in math but in life. It gives you confidence in your answer and helps you catch any mistakes you might have made along the way. So, always remember to double-check your work – your future self will thank you for it!
Common Mistakes and How to Avoid Them
Solving equations can be tricky, and it's easy to make mistakes if you're not careful. One common mistake is forgetting to apply the same operation to both sides of the equation. Remember, the equation is like a balance scale, and whatever you do to one side, you must do to the other to keep it balanced. Another mistake is messing up the order of operations. Always remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Make sure you're performing operations in the correct order. A really common slip-up is with the signs – negative signs can be sneaky! Double-check your addition and subtraction, especially when dealing with negative numbers. And finally, don't skip the verification step! Plugging your solution back into the original equation is the best way to catch errors. To avoid these mistakes, practice is key. The more you solve equations, the more comfortable you'll become with the process. Take your time, show your work, and double-check each step. If you're struggling, don't be afraid to ask for help – your teacher, classmates, or even online resources can be great sources of support. With a little practice and attention to detail, you'll be solving equations like a pro!
Practice Problems: Sharpen Your Skills
Alright, now that we've gone through the steps and covered the common pitfalls, it's time to put your newfound skills to the test! The best way to master solving equations is to practice, practice, practice. Here are a few practice problems for you to try. Remember to follow the steps we discussed: isolate the variable term, solve for the variable, and verify your solution. Don't rush, take your time, and show your work. This will help you track your progress and identify any areas where you might need a little more help. And most importantly, don't be afraid to make mistakes – mistakes are how we learn! So, grab a pencil and paper, and let's get those brains working:
- 2x + 5 = 15
- -3y - 7 = 8
- 4 + 5z = -11
For each equation, work through the steps carefully. Pay attention to signs, remember to perform the same operation on both sides, and verify your answers. If you get stuck, go back and review the steps we discussed earlier. You can also look up examples online or ask a friend or teacher for help. The goal is not just to get the right answers but to understand the process. Solving equations is a fundamental skill in math, and it's one that you'll use again and again in your math journey. So, embrace the challenge, and enjoy the process of unlocking those solutions! With each equation you solve, you're building your confidence and strengthening your mathematical muscles.
Conclusion: You've Got This!
So there you have it! We've walked through the process of solving the linear equation 10 - 6v = -104 step-by-step, from understanding the equation to verifying our solution. You've learned how to isolate the variable term, solve for the variable, and avoid common mistakes. But more importantly, you've gained a valuable skill that will help you in all areas of math and beyond. Remember, solving equations is like learning any new skill – it takes practice, patience, and a willingness to learn from your mistakes. Don't get discouraged if you don't get it right away. Keep practicing, keep asking questions, and keep challenging yourself. With each equation you solve, you're building your problem-solving skills and your confidence in your abilities. You've got this! Now go out there and conquer those equations! And remember, math can be fun, especially when you start to see the patterns and understand the logic behind it. Keep exploring, keep learning, and keep growing your mathematical mind! You're amazing, and you're capable of achieving anything you set your mind to. So, keep shining and keep solving!
