Solving Equations: 4(t+1) = 6t-1 Explained

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Solving Equations: 4(t+1) = 6t-1 Explained

Hey there, math enthusiasts! Today, we're diving into the equation 4(t+1) = 6t - 1. Don't worry, it looks more intimidating than it actually is. We're going to break down this problem step-by-step to find the value of 't' that makes this equation true. Think of it like a puzzle; our goal is to isolate 't' on one side of the equation and reveal its secret value. We'll use some basic algebraic principles, and I promise, it's not as scary as it might seem at first glance. We'll go through each step, making sure you understand the 'why' behind every move. By the end, you'll be able to confidently tackle similar equations. So, let's get started and unravel this mathematical mystery together! This is the core of algebra and understanding how to solve these kinds of equations is essential for success in higher-level math. We'll show you the straightforward approach to not only solve for 't' but also gain a deeper understanding of how the different parts of the equation relate to each other. Get ready to flex those brain muscles and have some fun with equations!

Step-by-Step Solution

Alright, let's get our hands dirty and actually solve this equation. The key to solving 4(t+1) = 6t - 1 is to methodically simplify the equation until we get 't' all by itself on one side. Remember, the goal is to find the value of 't' that makes the equation balanced. Here’s a breakdown of the steps:

  1. Distribute the 4: The first step is to get rid of those parentheses. We do this by distributing the 4 across the terms inside the parentheses. So, 4 multiplies by 't' and also by 1. This gives us 4t + 41, which simplifies to 4t + 4. Now our equation looks like this: 4t + 4 = 6t - 1.

  2. Combine 't' terms: Next, we need to bring all the 't' terms together. To do this, we can subtract 4t from both sides of the equation. This gets rid of the '4t' on the left side. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. After subtracting 4t from both sides, our equation becomes: 4 = 2t - 1.

  3. Isolate the constant terms: Now, let's get the constant terms (the numbers without 't') together. We add 1 to both sides of the equation to get rid of the '-1' on the right side. This gives us: 5 = 2t.

  4. Solve for 't': Finally, we need to isolate 't'. To do this, we divide both sides of the equation by 2. This leaves us with: t = 5/2. Converting this improper fraction to a mixed number, we get t = 2 1/2. And there you have it, folks! We've solved for 't'! Each step is designed to simplify the equation, making it easier to see how 't' relates to the other numbers. Remember, practice makes perfect, and the more you work through these problems, the more confident you'll become.

Why Each Step Matters

Let's pause and really understand why we took each step, and what algebraic principles were at play. Each move wasn't random; they are grounded in the core principles of algebra. When we distributed the 4, we were using the distributive property, which states that a(b + c) = ab + ac. This property is crucial for simplifying expressions containing parentheses. Then, when we combined the 't' terms and the constants, we used the additive and subtractive properties of equality. These properties say that if you add or subtract the same value from both sides of an equation, the equation remains balanced. It's like a seesaw; to keep it balanced, you have to add or remove weight from both sides simultaneously. Understanding these principles helps to transform complex-looking equations into something manageable. They're the rules of the game, and once you get them, solving equations becomes much easier and more intuitive. Each step works together, each action we take is meant to isolate 't', making the equation more clear and making it easy to determine the value of 't'.

Checking Your Answer

Alright, you've solved the equation, great job! But hold up—it's super important to double-check your work. How do you know if your answer is correct? Simple: plug the value of 't' back into the original equation. This is called verifying your solution. Let's substitute t = 2 1/2 (or 5/2) into 4(t+1) = 6t - 1. This is the process:

  • Original equation: 4(t + 1) = 6t - 1.
  • Substitute t = 5/2: 4(5/2 + 1) = 6(5/2) - 1.
  • Simplify: 4(7/2) = 15 - 1.
  • Continue simplifying: 14 = 14.

See? The equation holds true! Since both sides equal each other after plugging in our value for 't', it's safe to say we've nailed the correct answer. Verifying your solution is a great habit to cultivate. It not only confirms your answer but also helps you to catch any potential errors you might have made along the way. Think of it as a safety net that boosts your confidence in the accuracy of your solutions. This simple step can save you from a lot of frustration down the road. It reinforces your understanding and reinforces that you have correctly applied all the required steps. Always remember to check your work; it's a mark of a diligent mathematician!

Common Mistakes to Avoid

As you're navigating the world of algebra, here are some common pitfalls that students often stumble upon when solving equations like 4(t+1) = 6t - 1. Knowing these traps ahead of time can help you avoid making the same mistakes and improve your overall problem-solving skills.

  • Incorrect Distribution: One of the most common mistakes is improperly distributing. Forgetting to multiply all terms inside the parentheses by the factor outside is a big one. For example, when simplifying 4(t+1), some might only multiply the 't' but miss multiplying the 1. Always make sure to distribute to every single term inside. Double-check your distribution as the first step.

  • Sign Errors: Pay close attention to the signs (plus or minus) throughout the equation. A small slip-up in signs can completely change your answer. For example, misinterpreting -1 as +1 can lead to a wrong solution. Always make sure to check and double-check all signs, especially when moving terms across the equals sign. A simple way to do this is to rewrite the signs on a separate piece of paper.

  • Combining Unlike Terms: This happens when you try to add or subtract terms that aren't similar. For instance, you can't combine a 't' term with a constant term directly. Remember, you can only add or subtract terms that have the same variable and exponent. These errors are easy to make when you are learning and can be improved with practice. Therefore, it is important to practice regularly, to see the different types of problems and how to solve them.

Conclusion

So there you have it, folks! We've successfully solved 4(t+1) = 6t - 1, and the answer is t = 2 1/2. We've walked through the step-by-step solution, explained the 'why' behind each step, and learned how to verify our answer to ensure accuracy. Remember, solving equations is all about breaking down complex problems into smaller, manageable steps. By consistently applying the rules of algebra and practicing regularly, you'll become more and more comfortable with solving a variety of equations. Keep practicing, stay curious, and don't be afraid to make mistakes—they're all part of the learning process! Keep going, and you'll become a pro at these problems in no time! Keep practicing, and you'll become a pro at these problems in no time! So, keep exploring the world of math, and remember that every equation you solve brings you one step closer to mastering algebra.