Solving For T: A Step-by-Step Guide To $-3t + 1 = T - 11$

Hey guys! Let's dive into solving a simple algebraic equation today. We're going to tackle the equation -3t + 1 = t - 11 and find out what value of t makes this statement true. This is a fundamental skill in algebra, and by the end of this guide, you'll be a pro at solving similar equations. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into the solution, it’s important to really understand what the equation -3t + 1 = t - 11 is telling us. Think of an equation as a balanced scale. The left side of the equals sign has the same value as the right side. Our goal is to isolate t on one side of the equation to figure out what numerical value it represents. In this particular equation, t is our variable, and we need to find the specific value for t that maintains the balance between both sides. Remember, whatever we do to one side of the equation, we must also do to the other to keep things balanced. This principle is crucial in solving any algebraic equation. So, let's keep this idea of balance in mind as we move forward and start working through the steps to isolate t. By understanding the structure of the equation, we set ourselves up for a smoother and more intuitive solving process. The key to success in algebra often lies in thoroughly understanding the basics.

Step-by-Step Solution

Alright, let's break down how to solve for t in the equation -3t + 1 = t - 11 step-by-step. We’ll go through each action and explain why we're doing it, so you get a clear picture of the process. First things first, our main goal here is to get all the t terms on one side of the equation and the constant terms (the numbers) on the other side. This will help us isolate t and eventually find its value. Remember, the principle of maintaining balance is key – what we do to one side, we must do to the other.

Step 1: Combining Like Terms

The first move we're going to make is to collect all the t terms on one side of the equation. A common way to do this is to add 3t to both sides of the equation. Why? Because adding 3t to the left side will cancel out the -3t that's already there, effectively moving the t term to the right side. It looks like this:

-3t + 1 + 3t = t - 11 + 3t

When we simplify this, the -3t and +3t on the left side cancel each other out, leaving us with just 1. On the right side, t and 3t combine to give 4t. So, the equation now looks like this:

1 = 4t - 11

Now, we have all our t terms on the right side, which is exactly what we wanted! This step is all about making the equation simpler and easier to work with. By strategically adding 3t to both sides, we've taken a big step toward isolating our variable, t.

Step 2: Isolating the Variable Term

Now that we've grouped the t terms together, the next goal is to isolate the term that contains t. In our current equation, 1 = 4t - 11, the term we want to isolate is 4t. To do this, we need to get rid of the -11 that's on the same side of the equation. The way we do that is by adding 11 to both sides. Remember the balance – whatever we do to one side, we have to do to the other. Adding 11 to both sides looks like this:

1 + 11 = 4t - 11 + 11

On the left side, 1 + 11 simplifies to 12. On the right side, -11 + 11 cancels out, leaving us with just 4t. So, our equation now becomes:

12 = 4t

Great! We've successfully isolated the term with t. This step is crucial because now we're just one step away from finding the value of t itself. By adding 11 to both sides, we've simplified the equation and brought ourselves closer to the final solution. Let's move on to the last step and solve for t!

Step 3: Solving for t

We've made it to the final step! We're at the equation 12 = 4t, and our mission now is to get t all by itself on one side. Currently, t is being multiplied by 4. To undo this multiplication, we need to do the opposite operation, which is division. So, we're going to divide both sides of the equation by 4. Again, we must do the same thing to both sides to maintain balance.

Dividing both sides by 4 looks like this:

12 / 4 = (4t) / 4

On the left side, 12 divided by 4 is 3. On the right side, 4t divided by 4 simplifies to just t. This is exactly what we wanted! We now have t isolated on one side of the equation. So, the equation simplifies to:

3 = t

Or, we can write it as:

t = 3

And there you have it! We've successfully solved for t. This final step of dividing both sides by the coefficient of t is a common technique in algebra. It's the last piece of the puzzle that allows us to reveal the value of the variable. So, in this case, t equals 3. Pat yourself on the back – you've just solved an algebraic equation!

Verifying the Solution

Okay, we've arrived at a solution, t = 3, but it's always a good idea to double-check our work to make sure we didn't make any sneaky errors along the way. Verifying our solution gives us peace of mind and ensures that our answer is correct. To do this, we're going to substitute t = 3 back into the original equation, -3t + 1 = t - 11, and see if both sides of the equation end up being equal. If they do, we know our solution is solid. If they don't, we'll need to go back and check our steps.

So, let's plug in t = 3:

-3(3) + 1 = (3) - 11

Now, we simplify each side independently. On the left side, -3 multiplied by 3 is -9, so we have:

-9 + 1 = 3 - 11

Simplifying further, -9 + 1 equals -8. On the right side, 3 - 11 also equals -8. So, our equation now looks like this:

-8 = -8

Voila! Both sides of the equation are equal. This confirms that our solution, t = 3, is indeed correct. Verifying the solution is a crucial step in problem-solving. It’s like the final seal of approval on our work. By substituting the value back into the original equation, we ensure that our answer maintains the balance and satisfies the equation. So, always take that extra step to verify – it's worth it!

Final Answer

Alright guys, after our step-by-step journey through the equation -3t + 1 = t - 11, we've not only found the solution but also verified it. We combined like terms, isolated the variable term, solved for t, and then double-checked our work. So, drumroll please…

The final answer is:

t = 3

That’s it! We’ve successfully navigated this algebraic puzzle and emerged victorious. Give yourself a pat on the back – you've earned it! Remember, solving equations is a fundamental skill in mathematics, and mastering it opens doors to more complex concepts. So, keep practicing, keep exploring, and keep that mathematical curiosity alive. You’ve got this!