Solving 5x + 18 = 8 A Step-by-Step Guide And Verification

by Kenji Nakamura 58 views

Hey guys! Today, we're going to break down how to solve and verify the equation 5x + 18 = 8. Math can seem daunting, but trust me, with a step-by-step approach, it becomes super manageable. We'll go through each stage, ensuring you understand not just the how, but also the why behind every move. Let’s dive in and make math a little less mysterious, shall we?

Understanding the Basics

Before we jump into the problem, it's crucial to understand the basic principles we'll be using. Solving an equation means finding the value of the unknown variable (in this case, x) that makes the equation true. To do this, we use inverse operations to isolate the variable on one side of the equation. Think of an equation like a balance scale; whatever you do to one side, you must do to the other to keep it balanced. The key concepts here are inverse operations and maintaining the balance of the equation. For instance, the inverse operation of addition is subtraction, and the inverse of multiplication is division. We'll apply these concepts methodically to solve for x in our equation. The foundation of algebra rests on these principles, and grasping them will empower you to tackle more complex problems down the road. So, before we get into the nitty-gritty, remember that the goal is to isolate x by using operations that 'undo' what's being done to it, always ensuring we treat both sides of the equation equally. This balanced approach is the golden rule of equation solving, and it’s what will guide us to the correct solution every time.

Step 1: Isolating the Term with 'x'

Okay, so we've got 5x + 18 = 8. Our first goal is to get the term with x (which is 5x) all by itself on one side of the equation. To do this, we need to get rid of the + 18. How do we do that? Well, we use the inverse operation. The opposite of adding 18 is subtracting 18. Remember, whatever we do to one side of the equation, we have to do to the other. So, we subtract 18 from both sides:

  • 5x + 18 - 18 = 8 - 18

This simplifies to:

  • 5x = -10

See what we did there? By subtracting 18 from both sides, we effectively canceled out the +18 on the left side, leaving us with just 5x. This is a crucial step because it brings us closer to isolating x. Think of it as peeling away the layers of an onion; we’re systematically removing everything around x until it stands alone. This process of isolating terms is a fundamental technique in algebra, and mastering it will make solving equations significantly easier. The clarity and precision we apply in this initial step set the stage for the rest of the solution, ensuring we maintain accuracy and avoid common pitfalls. So, let’s take a moment to appreciate the power of inverse operations in simplifying our equation and setting us up for success.

Step 2: Solving for 'x'

Now that we've isolated 5x, we have the equation 5x = -10. Our next step is to get x completely by itself. Currently, x is being multiplied by 5. To undo this multiplication, we need to use the inverse operation, which is division. We'll divide both sides of the equation by 5:

  • (5x) / 5 = (-10) / 5

This simplifies to:

  • x = -2

And there you have it! We've solved for x. It turns out that x equals -2. This step is where the magic really happens; it's where we finally reveal the value of our unknown variable. The division we performed effectively 'undoes' the multiplication, allowing x to stand alone and reveal its true identity. This process highlights the elegance of algebraic manipulation, where carefully chosen operations lead us to a clear and concise solution. Remember, each step we take is deliberate, aimed at simplifying the equation and bringing us closer to our goal. Solving for x is not just about getting a number; it's about understanding the relationship between the variable and the constants in the equation. With this understanding, you can confidently tackle a wide range of algebraic problems and appreciate the power of mathematics in uncovering hidden values.

Step 3: Verifying the Solution

Okay, we've found that x = -2, but how do we know if we're right? This is where verification comes in. It's like a final check to make sure our answer makes sense. To verify our solution, we substitute the value of x back into the original equation:

  • Original equation: 5x + 18 = 8
  • Substitute x = -2: 5(-2) + 18 = 8

Now, let's simplify:

  • -10 + 18 = 8
  • 8 = 8

See? The equation holds true! This means our solution, x = -2, is correct. Verification is a crucial step in problem-solving because it provides concrete evidence that our solution is accurate. It's like the final piece of a puzzle clicking into place, giving us confidence in our work. By substituting our calculated value back into the original equation, we’re essentially testing whether our solution maintains the equation's balance. If both sides of the equation are equal after the substitution, we know we’re on the right track. This process not only confirms our answer but also deepens our understanding of the equation itself. Verification reinforces the idea that mathematics is not just about finding numbers; it's about ensuring those numbers fit within the logical framework of the problem. So, always take the time to verify your solutions – it’s a surefire way to boost your confidence and avoid simple mistakes.

Common Mistakes to Avoid

When solving equations, there are a few common pitfalls that students often stumble into. Let's highlight some of these so you can steer clear of them:

  1. Forgetting to apply operations to both sides: This is a big one! Remember the balance scale analogy? Whatever you do to one side of the equation, you must do to the other. If you subtract 18 from the left side, you have to subtract it from the right side too. Failing to do so will throw off the entire equation and lead to an incorrect answer. It’s like trying to build a balanced structure with uneven foundations – it’s simply not going to work. Consistency in applying operations is the bedrock of equation solving, so always double-check that you’re treating both sides equally. This simple habit can save you a lot of headaches and ensure your solutions are accurate.

  2. Incorrectly applying the order of operations: Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? It's crucial to follow this order when simplifying expressions. Messing up the order can lead to incorrect calculations. For example, if you add before you multiply, you’ll end up with the wrong result. Think of PEMDAS as the recipe for solving equations; follow the steps in the right order, and you’ll get a delicious (and correct) answer. Paying close attention to the order of operations may seem tedious at first, but it’s a skill that will pay off immensely as you tackle more complex mathematical problems. It’s the key to maintaining accuracy and ensuring your solutions stand up to scrutiny.

  3. Sign errors: Watch out for those pesky positive and negative signs! A simple sign error can completely change the outcome of the problem. Always double-check your signs, especially when dealing with subtraction and negative numbers. It’s easy to overlook a minus sign or misinterpret a subtraction as an addition, but these small slips can have big consequences. Consider each sign a critical piece of information, and take the time to verify that it’s correct. This attention to detail is a hallmark of careful problem-solving and can significantly reduce the likelihood of errors. Think of sign errors as the gremlins of mathematics – they’re small and easily missed, but they can wreak havoc if left unchecked. So, be vigilant and keep those signs in line!

  4. Not verifying the solution: As we discussed earlier, verification is key. Don't skip this step! Plugging your answer back into the original equation is the best way to catch any mistakes. It’s like proofreading a document before you submit it; it’s your chance to catch any errors and make sure everything is perfect. Verification is more than just a formality; it’s a powerful tool for building confidence in your work. It provides concrete evidence that your solution is correct and reinforces your understanding of the equation. So, make it a habit to verify every solution – it’s a small investment of time that can yield big rewards in terms of accuracy and understanding.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving equations.

Conclusion

So, guys, we've successfully solved and verified the equation 5x + 18 = 8! We walked through each step, from isolating the term with x to the final verification. Remember, math is like a puzzle, and each step is a piece that fits together to reveal the solution. By understanding the basic principles and avoiding common mistakes, you can tackle any equation with confidence. Keep practicing, and you'll become a math whiz in no time! The journey of learning mathematics is like climbing a staircase; each step you take builds upon the previous one, leading you to a higher level of understanding. Equations are fundamental building blocks in this journey, and mastering them opens doors to more complex and fascinating mathematical concepts. Embrace the challenge, stay curious, and never hesitate to break down problems into smaller, more manageable steps. With patience and persistence, you’ll not only solve equations but also develop a deeper appreciation for the beauty and logic of mathematics. So, keep practicing, keep learning, and keep exploring the wonderful world of numbers and symbols – you’ll be amazed at what you can achieve!