Solving For A The Equation -9a + 4 = 112 A Step-by-Step Guide

Hey guys! Are you struggling with solving algebraic equations? Don't worry, we've all been there. In this article, we're going to break down a common type of problem: solving for a variable. Specifically, we'll be tackling the equation -9a + 4 = 112. This might seem intimidating at first, but with a step-by-step approach, you'll be solving for 'a' like a pro in no time! We'll cover each step in detail, explaining the logic behind it so you not only get the answer but also understand the why behind the math. So, grab your pencils, and let's dive in!

Understanding the Basics of Algebraic Equations

Before we jump into solving the equation, let's quickly recap some fundamental concepts of algebraic equations. Algebraic equations are mathematical statements that show the equality between two expressions. They typically involve variables (like 'a' in our case), constants (numbers like 4 and 112), and mathematical operations (addition, subtraction, multiplication, division). The goal of solving an equation is to isolate the variable on one side of the equation to determine its value. Think of it like a puzzle – we need to manipulate the equation to reveal the hidden value of our variable. This manipulation involves using inverse operations. Each operation has its inverse: addition's inverse is subtraction, subtraction's inverse is addition, multiplication's inverse is division, and division's inverse is multiplication. We use these inverse operations to "undo" the operations affecting our variable, gradually isolating it. The golden rule in solving equations is to maintain balance. What you do to one side of the equation, you must do to the other side. This ensures the equality remains valid throughout the solving process. Keeping this in mind will help prevent common errors and lead you to the correct solution. Now that we've refreshed the basics, we are totally ready to solve the equation!

Step-by-Step Solution for -9a + 4 = 112

Okay, let's get our hands dirty and solve the equation -9a + 4 = 112 step-by-step. Remember our goal: to isolate 'a' on one side of the equation.

Step 1: Isolate the Term with the Variable

The first thing we need to do is get the term containing 'a' (-9a) by itself on one side of the equation. To do this, we need to eliminate the constant term (+4) that's being added to it. The inverse operation of addition is subtraction, so we'll subtract 4 from both sides of the equation. This is super important – whatever we do to one side, we must do to the other to keep the equation balanced. So, we have:

-9a + 4 - 4 = 112 - 4

This simplifies to:

-9a = 108

Great! We've successfully isolated the term with 'a'.

Step 2: Isolate the Variable

Now, 'a' is being multiplied by -9. To isolate 'a', we need to undo this multiplication. The inverse operation of multiplication is division, so we'll divide both sides of the equation by -9:

-9a / -9 = 108 / -9

This simplifies to:

a = -12

And there you have it! We've solved for 'a'.

Step 3: Verify the Solution

It's always a good idea to check your answer to make sure it's correct. To do this, we'll substitute our solution (a = -12) back into the original equation:

-9(-12) + 4 = 112

Let's simplify:

108 + 4 = 112

112 = 112

Our solution checks out! Both sides of the equation are equal, which means a = -12 is indeed the correct answer. We have successfully navigated the process of solving for variables in an algebraic equation, demonstrating the importance of inverse operations and maintaining balance in the equation. Each step was carefully executed, ensuring accuracy and understanding.

Common Mistakes to Avoid When Solving Equations

Solving equations can be tricky, and it's easy to make mistakes if you're not careful. Let's go over some common pitfalls to avoid. One frequent error is failing to perform the same operation on both sides of the equation. Remember, maintaining balance is crucial. If you subtract a number from one side, you must subtract it from the other. Another common mistake involves incorrectly applying the order of operations (PEMDAS/BODMAS). Make sure you're performing operations in the correct order (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Sign errors are also a big one. Pay close attention to negative signs, especially when dividing or multiplying. A misplaced negative sign can completely change your answer. Forgetting to distribute is another common issue. If you have a number multiplied by a term in parentheses, make sure you distribute it to every term inside the parentheses. Finally, always double-check your work! It's easy to make a small arithmetic error, so taking a few extra seconds to review your steps can save you a lot of headaches. By being aware of these common mistakes, you can significantly improve your accuracy and confidence in solving equations. These errors are often the result of rushing through the problem or overlooking small details, but with practice and attention to detail, you can minimize these mistakes and strengthen your problem-solving skills.

Practice Problems to Sharpen Your Skills

Now that you've learned how to solve for a variable, it's time to put your skills to the test with some practice problems. Practice is key to mastering any mathematical concept, and solving equations is no exception. The more problems you solve, the more comfortable and confident you'll become. So, let’s test your skills with a few more examples:

1. Solve for x: 5x - 10 = 25
2. Solve for y: -3y + 7 = -8
3. Solve for z: 2z + 15 = 31

Pro Tip: Remember to show your work for each problem! This will help you identify any errors you might be making and make it easier to learn from your mistakes. Once you have your solutions, check them by substituting them back into the original equations, just like we did earlier. This is the best way to ensure you've got the correct answer. Treat each problem as a small challenge and a chance to reinforce what you’ve learned. The goal isn’t just to find the answer, but to understand the process. Embrace the challenge, and remember that every mistake is a learning opportunity. Keep practicing, and you’ll find that solving equations becomes second nature. And the more you practice, the more proficient you'll become in navigating algebraic challenges.

Conclusion: You've Got This!

Alright guys, we've covered a lot in this article! We started with the basics of algebraic equations, walked through a step-by-step solution for -9a + 4 = 112, discussed common mistakes to avoid, and even gave you some practice problems to try. Remember, solving equations is a skill that gets better with practice. Don't get discouraged if you don't get it right away. The key is to understand the underlying concepts and to approach each problem systematically. Use the techniques we discussed, pay attention to detail, and always double-check your work. You've learned the importance of inverse operations in undoing mathematical actions and the critical role of maintaining balance in the equation. Each step in solving equations is a careful maneuver to isolate the variable, revealing its true value. With every equation you solve, you're building your problem-solving muscles and deepening your understanding of algebra. So, keep practicing, stay curious, and you'll be solving even the most complex equations with confidence in no time. You've got this!