Solve 2(x+10)=50: Step-by-Step Guide

Hey guys! Today, we're diving into a common type of math problem: solving linear equations. Specifically, we're going to tackle the equation 2(x + 10) = 50. Don't worry if it looks a bit intimidating at first. We'll break it down step-by-step, so you'll be solving equations like a pro in no time! Whether you're a student brushing up on your algebra skills or just someone who enjoys a good math challenge, this guide is for you. We'll not only show you how to solve the equation but also explain the why behind each step, making sure you understand the underlying principles. So, grab your pencil and paper, and let's get started!

Understanding the Basics of Linear Equations

Before we jump into the solution, let's make sure we're all on the same page about what a linear equation is and the basic principles involved in solving them. Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. The variable is only raised to the first power (no exponents like x² or x³). These equations, when graphed, produce a straight line – hence the name “linear.” Think of it like a balancing scale. Our goal is to isolate the variable (in this case, 'x') on one side of the equation, just like balancing the scale so that only 'x' remains on one side. To do this, we use inverse operations. This means we perform the opposite operation to both sides of the equation to maintain the balance. For example, if we have addition, we use subtraction; if we have multiplication, we use division, and vice versa. Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. This is the golden rule of equation solving! This concept is crucial because it ensures that the equality remains valid throughout the process. Each step we take is designed to simplify the equation while preserving its original meaning. By understanding this fundamental principle, you'll be well-equipped to solve a wide range of linear equations with confidence. Now that we've covered the basics, let's move on to the specific steps involved in solving our equation, 2(x + 10) = 50.

Step 1: Distribute the 2

Okay, the first thing we need to do in our equation, 2(x + 10) = 50, is to get rid of those parentheses. To do this, we'll use the distributive property. Remember the distributive property? It basically says that a number multiplied by a sum (or difference) inside parentheses is the same as multiplying the number by each term inside the parentheses separately. In our case, we have 2 multiplied by (x + 10). So, we need to distribute the 2 to both the 'x' and the '10'. This means we multiply 2 by 'x', which gives us 2x, and then we multiply 2 by 10, which gives us 20. So, 2(x + 10) becomes 2x + 20. Now, we rewrite the entire equation with this simplification. Our equation now looks like this: 2x + 20 = 50. See how much cleaner that looks already? Distributing the number outside the parenthesis is a crucial step because it allows us to separate the terms and start isolating our variable, 'x'. Without distributing, we wouldn't be able to combine like terms or perform other operations necessary to solve for 'x'. This step is like unlocking the door to the rest of the solution. It sets the stage for the subsequent steps and brings us closer to our final answer. So, always remember to look for parentheses and apply the distributive property first – it's a key tool in your equation-solving arsenal!

Step 2: Subtract 20 from Both Sides

Alright, we've distributed the 2, and our equation now looks like 2x + 20 = 50. The next step is to isolate the term with 'x' on one side of the equation. Currently, we have a '+ 20' hanging out with the '2x'. To get rid of it, we need to perform the inverse operation. The inverse of addition is subtraction, so we're going to subtract 20 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced! So, we subtract 20 from the left side (2x + 20 - 20) and we subtract 20 from the right side (50 - 20). On the left side, the +20 and -20 cancel each other out, leaving us with just 2x. On the right side, 50 - 20 equals 30. Our equation now simplifies to 2x = 30. See how we're getting closer to isolating 'x'? Subtracting 20 from both sides was a crucial step in unwrapping the equation. It's like peeling away a layer to reveal the variable we're trying to solve for. By performing this step, we've effectively moved the constant term to the other side of the equation, paving the way for the final step where we isolate 'x' completely. This highlights the importance of inverse operations in equation solving – they're the key to moving terms around and simplifying the equation.

Step 3: Divide Both Sides by 2

Okay, we're almost there! Our equation is currently 2x = 30. We have '2x', which means 2 multiplied by 'x'. To finally isolate 'x', we need to undo this multiplication. The inverse operation of multiplication is division, so we're going to divide both sides of the equation by 2. Again, remember the golden rule: what we do to one side, we must do to the other! So, we divide the left side (2x) by 2, and we divide the right side (30) by 2. On the left side, 2x divided by 2 simplifies to just 'x'. On the right side, 30 divided by 2 equals 15. This gives us our final solution: x = 15. Hooray! We've solved the equation! Dividing both sides by 2 was the final step in isolating 'x' and finding its value. It's like the last piece of the puzzle falling into place. This step underscores the power of using inverse operations to undo mathematical operations and reveal the solution. By dividing, we effectively separated 'x' from its coefficient, leaving us with the value of 'x' itself. This is the ultimate goal in solving any equation – to get the variable all by itself on one side, so we know its value. Now that we've found x = 15, let's take a moment to check our answer and make sure it's correct.

Step 4: Check Your Answer

It's always a good idea to check your answer, especially in math! This helps you catch any mistakes and gives you confidence that your solution is correct. So, let's take our solution, x = 15, and plug it back into the original equation, 2(x + 10) = 50, to see if it holds true. We'll replace 'x' with 15: 2(15 + 10) = 50. Now, we need to simplify the left side of the equation. First, we solve what's inside the parentheses: 15 + 10 = 25. So, our equation now looks like this: 2(25) = 50. Next, we multiply 2 by 25, which gives us 50. So, we have 50 = 50. This is a true statement! Since the left side of the equation equals the right side when we substitute x = 15, we know that our solution is correct. Checking your answer is like having a safety net. It ensures that you haven't made any errors along the way and that your solution is valid. It's a simple step that can save you a lot of trouble in the long run, especially on tests or assignments. Plus, it feels great to confirm that you've solved the problem correctly! So, always make it a habit to check your answer after you've solved an equation – it's a smart and effective way to boost your confidence and accuracy.

Conclusion: You Did It!

Awesome job, guys! We've successfully solved the equation 2(x + 10) = 50 step-by-step. We distributed, subtracted, divided, and even checked our answer to make sure it was correct. You've learned how to tackle this type of linear equation, and the skills you've gained here will help you solve many more math problems in the future. Remember, the key to solving equations is to break them down into smaller, manageable steps and to use inverse operations to isolate the variable. Don't be afraid to take your time and work through each step carefully. And most importantly, practice makes perfect! The more you practice solving equations, the more comfortable and confident you'll become. So, keep practicing, keep learning, and keep challenging yourself. You've got this! Solving equations is a fundamental skill in algebra and beyond, so mastering it now will set you up for success in future math courses and in various real-world applications. Whether you're calculating finances, designing structures, or analyzing data, the ability to solve equations is a valuable asset. So, pat yourselves on the back for your hard work and dedication. You've added another valuable tool to your mathematical toolkit. Keep up the great work, and who knows what other mathematical challenges you'll conquer next! Now go forth and solve some more equations!