Solve 2(t+1)=10: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a classic algebraic problem: solving for the variable t in the equation 2(t + 1) = 10. Don't worry if algebra feels like a tangled mess right now; we'll break it down step-by-step, making it crystal clear. Whether you're a student tackling homework or just brushing up on your math skills, this guide will equip you with the tools to confidently solve similar equations.

Understanding the Equation: Our Starting Point

Before we jump into the solution, let's take a moment to understand what the equation 2(t + 1) = 10 is telling us. In essence, it's a mathematical statement that declares two expressions are equal. The left side, 2(t + 1), involves a variable (t) that we're trying to find. The right side is simply the number 10. Our mission is to isolate t on one side of the equation, revealing its value. This is a fundamental concept in algebra, and mastering it opens the door to solving more complex problems. So, let's embark on this algebraic adventure together and see how we can crack this equation. Remember, each step we take is a step closer to understanding the beauty and logic of mathematics!

The Distributive Property: Unlocking the Parentheses

Our first key move involves the distributive property. This nifty rule allows us to eliminate the parentheses in the equation. The distributive property states that a(b + c) = ab + ac. In simpler terms, we multiply the term outside the parentheses (in our case, 2) by each term inside the parentheses (t and 1). Applying this to our equation, 2(t + 1) becomes 2 * t + 2 * 1, which simplifies to 2t + 2. So, our equation now looks like this: 2t + 2 = 10. See how much cleaner that looks already? We've successfully untangled the expression, making it easier to work with. This step is crucial because it separates the variable t from the addition operation, paving the way for us to isolate it. Mastering the distributive property is like having a secret weapon in your algebraic arsenal, ready to be deployed whenever parentheses stand in your way. So, embrace this powerful tool, and let's move on to the next step in our journey to solve for t*.

Isolating the Term with 't': The Subtraction Step

Now that we've applied the distributive property, our equation stands at 2t + 2 = 10. Our next goal is to isolate the term containing t (which is 2t) on one side of the equation. To do this, we need to get rid of the '+ 2' that's hanging out on the left side. The golden rule of equation solving is that whatever operation you perform on one side, you must perform on the other to maintain balance. So, to eliminate the '+ 2', we'll subtract 2 from both sides of the equation. This gives us: 2t + 2 - 2 = 10 - 2. Simplifying this, we get 2t = 8. Awesome! We've successfully isolated the term with t. It's like we're peeling away the layers of the equation, revealing the hidden value of t step by step. This subtraction step is a classic technique in algebra, and you'll find yourself using it frequently. Remember, the key is to maintain balance – like a mathematical see-saw – ensuring that both sides of the equation remain equal. Now that we've cleared the path, let's move on to the final step and uncover the value of t!

Solving for 't': The Division Finale

We've reached the final stretch! Our equation is now beautifully simplified to 2t = 8. The term 2t means '2 multiplied by t', and our ultimate goal is to find the value of t itself. To do this, we need to undo the multiplication. And how do we undo multiplication? With division, of course! Just like before, we'll apply the same operation to both sides of the equation to maintain that crucial balance. We'll divide both sides by 2, giving us: (2t) / 2 = 8 / 2. On the left side, the 2s cancel each other out, leaving us with just t. On the right side, 8 divided by 2 is 4. So, we arrive at our solution: t = 4. Hallelujah! We've cracked the code and discovered the value of t! This division step is the final flourish in our algebraic dance, bringing us to a satisfying conclusion. Remember, every step we've taken has been deliberate and logical, building upon the previous one. By mastering these techniques, you're not just solving equations; you're unlocking the power of algebraic thinking.

The Solution: t = 4

After carefully navigating the equation 2(t + 1) = 10, we've arrived at our final answer: t = 4. But before we celebrate our victory, let's take a moment to appreciate the journey. We started with a seemingly complex equation, but by applying the distributive property and the principles of maintaining balance through subtraction and division, we systematically peeled away the layers to reveal the value of t. This process is the essence of algebra – transforming the unknown into the known. The solution t = 4 tells us that if we substitute 4 for t in the original equation, the equation will hold true. It's like a puzzle piece perfectly fitting into its place. So, give yourself a pat on the back! You've not only solved for t, but you've also reinforced your understanding of fundamental algebraic concepts. Now, go forth and conquer more mathematical challenges with confidence! Remember, every equation is just another opportunity to practice your skills and sharpen your mind. Keep exploring, keep learning, and keep solving!

Checking Our Work: Ensuring Accuracy

In the world of mathematics, it's always a good practice to double-check our work. This ensures that we haven't made any sneaky errors along the way and that our solution is indeed correct. So, now that we've found t = 4, let's plug this value back into our original equation, 2(t + 1) = 10, and see if it holds true. Substituting t with 4, we get: 2(4 + 1) = 10. Now, let's simplify the left side of the equation. First, we solve the expression inside the parentheses: 4 + 1 = 5. So, we have 2(5) = 10. Next, we perform the multiplication: 2 * 5 = 10. And there you have it! We arrive at 10 = 10, which is a true statement. This confirms that our solution, t = 4, is indeed correct. Woohoo! We not only solved the equation but also verified our answer, showcasing our mathematical prowess. This step of checking our work is crucial, especially in exams or real-world applications where accuracy is paramount. So, remember to always take that extra minute to verify your solutions – it's the hallmark of a confident and skilled mathematician.

Practice Makes Perfect: Sharpening Your Skills

Congratulations on successfully solving for t! But remember, like any skill, mastering algebra requires practice. The more equations you solve, the more comfortable and confident you'll become. Think of it like learning a musical instrument – the more you practice, the smoother your fingers glide across the keys, and the more beautiful the music you create. Similarly, the more you practice solving algebraic equations, the more fluidly you'll navigate the steps, and the more elegant your solutions will become. So, don't be afraid to tackle a variety of problems, from simple to complex. Look for opportunities to apply your skills in different contexts. You might even find math puzzles and games that make practice more enjoyable. The key is to keep your mind engaged and to view challenges as opportunities for growth. Remember, every mistake is a learning opportunity, a chance to refine your understanding and strengthen your problem-solving abilities. So, embrace the practice, relish the challenges, and watch your algebraic skills soar! And hey, if you ever get stuck, don't hesitate to seek help from teachers, classmates, or online resources. The mathematical community is a supportive one, always ready to lend a hand. Now, go forth and conquer the world of equations!

Conclusion: The Power of Algebra

We've reached the end of our journey to solve for t in the equation 2(t + 1) = 10. We've not only found the solution, t = 4, but we've also explored the fundamental concepts of algebra that made it possible. From the distributive property to maintaining balance through operations on both sides of the equation, we've equipped ourselves with valuable tools for tackling similar problems. But more than just solving a single equation, we've gained a glimpse into the power of algebra. Algebra is more than just manipulating symbols and numbers; it's a way of thinking, a method for unraveling the unknown and revealing the underlying structure of mathematical relationships. It's a skill that extends far beyond the classroom, finding applications in science, engineering, economics, and countless other fields. So, embrace the power of algebra, and continue to explore its vast and fascinating landscape. Remember, every equation solved is a step forward on your mathematical journey, a step towards unlocking new knowledge and understanding. Keep practicing, keep exploring, and keep the spirit of mathematical inquiry alive! The world of mathematics awaits your discoveries.