Solving 5x + 6x - 16 = 33x + 2x - 4 A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving deep into solving a classic algebraic equation. Algebraic equations might seem daunting at first, but trust me, with a step-by-step approach, they become super manageable. Our mission today is to break down the equation 5x + 6x - 16 = 33x + 2x - 4. We'll tackle this by simplifying each side, combining like terms, and isolating our variable, 'x'. So, grab your pencils and let's get started! We aim to not only solve this particular problem but also to equip you with the skills to handle similar equations with confidence. Understanding the underlying principles is key, so we will focus on explaining each step clearly and thoroughly. Remember, math is like building blocks – each concept builds upon the previous one. By mastering these fundamental algebraic techniques, you’ll be well-prepared for more advanced topics in the future. Whether you're a student brushing up on your algebra skills or just curious about math, this guide is designed to be helpful and engaging. Let's transform this equation from a puzzling problem into a solved solution!

Step 1: Simplifying Both Sides of the Equation

Okay, first things first, let's simplify both sides of our equation. Our equation is 5x + 6x - 16 = 33x + 2x - 4. The secret here is to combine like terms. Think of it like organizing your closet – you put shirts with shirts, pants with pants, and so on. In math, we combine terms that have the same variable raised to the same power. On the left side, we have 5x and 6x. These are like terms because they both have 'x' to the power of 1. So, we simply add their coefficients (the numbers in front of 'x'). 5x + 6x equals 11x. The '-16' doesn't have an 'x', so it stays as it is. So, the left side simplifies to 11x - 16. Now, let's tackle the right side: 33x + 2x - 4. We see 33x and 2x – another pair of like terms. Adding them together, 33x + 2x equals 35x. And just like before, the '-4' remains unchanged since it’s a constant. This means the right side simplifies to 35x - 4. See? We've already made our equation look much cleaner: 11x - 16 = 35x - 4. By combining like terms, we’ve reduced the complexity and made the equation easier to work with. This step is crucial because it sets the stage for the next steps, where we'll be isolating 'x' to find its value. Remember, simplification is your friend in algebra!

Step 2: Moving Variables to One Side

Alright, now that we've simplified both sides, it's time to gather all our 'x' terms on one side of the equation. We've got 11x - 16 = 35x - 4. The goal here is to isolate 'x', and to do that, we need to get all the terms with 'x' onto the same side. It doesn't matter which side we choose, but it's often easier to move the smaller 'x' term to the side with the larger 'x' term. In our case, we have 11x on the left and 35x on the right. Since 11x is smaller than 35x, we'll move the 11x to the right side. How do we do that? We use the magic of inverse operations! To get rid of 11x on the left, we subtract 11x from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. So, we have: 11x - 16 - 11x = 35x - 4 - 11x. On the left, 11x - 11x cancels out, leaving us with just -16. On the right, we combine the like terms 35x and -11x, which gives us 24x. So, our equation now looks like this: -16 = 24x - 4. We're making great progress! By moving the variables to one side, we’ve further simplified the equation and brought ourselves closer to solving for 'x'. This step is a cornerstone of solving algebraic equations, and mastering it will make your algebra journey much smoother.

Step 3: Isolating the Variable Term

Fantastic! We're cruising along nicely. Our equation now stands at -16 = 24x - 4. Our next mission, should we choose to accept it (and we do!), is to isolate the term with our variable, which in this case is 24x. To do this, we need to get rid of the '-4' on the right side of the equation. Remember our magic tool, inverse operations? We're going to use it again! The inverse operation of subtraction is addition. So, to cancel out the '-4', we add 4 to both sides of the equation. This gives us: -16 + 4 = 24x - 4 + 4. On the left side, -16 + 4 equals -12. On the right side, -4 + 4 cancels out, leaving us with just 24x. So, our equation simplifies to: -12 = 24x. Look how much simpler it's become! We've successfully isolated the variable term. This is a huge step because now we’re just one step away from finding the value of 'x'. By isolating the variable term, we’ve cleared the path for the final operation that will reveal the solution. This step highlights the power of using inverse operations to manipulate equations and bring us closer to our goal.

Step 4: Solving for x

Drumroll, please! We've reached the final step in our equation-solving adventure. Our equation is now -12 = 24x. The ultimate goal is to find the value of 'x', which means we need to get 'x' all by itself on one side of the equation. Currently, 'x' is being multiplied by 24. To undo this multiplication, we use our trusty friend, the inverse operation. The inverse of multiplication is division. So, we divide both sides of the equation by 24. This gives us: -12 / 24 = 24x / 24. On the right side, 24x / 24 simplifies to just 'x', which is exactly what we want! On the left side, -12 / 24 simplifies to -1/2 or -0.5. So, our solution is: x = -1/2 or x = -0.5. Hooray! We've done it! We've successfully solved the equation. By dividing both sides by the coefficient of 'x', we were able to isolate 'x' and find its value. This final step is a testament to the power of inverse operations and their role in unraveling algebraic equations. Remember, the solution to an equation is the value that makes the equation true. In this case, if we substitute -1/2 for 'x' in the original equation, both sides will be equal.

Step 5: Checking the Solution (Optional but Recommended)

Alright, we've found our solution, x = -1/2, but how do we know for sure it's correct? This is where checking our solution comes in handy. It's like proofreading your work – it helps catch any potential errors. To check our solution, we substitute x = -1/2 back into our original equation: 5x + 6x - 16 = 33x + 2x - 4. Let's plug in -1/2 for every 'x': 5(-1/2) + 6(-1/2) - 16 = 33(-1/2) + 2(-1/2) - 4. Now, let's simplify each side. On the left: 5(-1/2) = -5/2, 6(-1/2) = -6/2 = -3. So, we have -5/2 - 3 - 16. Converting to a common denominator of 2, we get -5/2 - 6/2 - 32/2 = -43/2. On the right: 33(-1/2) = -33/2, 2(-1/2) = -1. So, we have -33/2 - 1 - 4. Converting to a common denominator of 2, we get -33/2 - 2/2 - 8/2 = -43/2. Guess what? Both sides are equal! -43/2 = -43/2. This confirms that our solution, x = -1/2, is indeed correct. Checking your solution is a great habit to develop. It not only ensures accuracy but also reinforces your understanding of the equation-solving process. It's like having a secret weapon against mistakes!

Conclusion

Woohoo! We did it! We successfully solved the equation 5x + 6x - 16 = 33x + 2x - 4, and we found that x = -1/2. But more importantly, we walked through the entire process step by step, from simplifying both sides to checking our solution. We covered combining like terms, using inverse operations, isolating the variable, and the importance of verifying our answer. These are fundamental skills in algebra, and mastering them will open doors to more complex mathematical concepts. Remember, solving equations is like solving a puzzle – each step brings you closer to the final answer. Don't be afraid to break down problems into smaller, more manageable steps. And most importantly, practice makes perfect! The more you practice, the more comfortable and confident you'll become in your equation-solving abilities. So, keep up the great work, and happy solving!