Solving The Equation: Find The Value Of T

Hey guys! Let's dive into solving this equation together. Equations might seem tricky at first, but with a step-by-step approach, we can break them down and find the solution. This article will guide you through the process of solving the equation $5(t-1)+3(t-3)=-2(t-1)+11(t+3)$. We'll explore each step in detail, making sure you understand the logic behind it. So, grab a pen and paper, and let's get started!

Understanding the Equation

Before we jump into the solution, let’s take a closer look at the equation: $5(t-1)+3(t-3)=-2(t-1)+11(t+3)$. Our main goal here is to find the value of the variable 't' that makes this equation true. To do this, we need to simplify the equation by expanding the parentheses, combining like terms, and isolating 't' on one side of the equation. This involves using the distributive property and basic algebraic operations like addition, subtraction, multiplication, and division. Remember, whatever we do to one side of the equation, we must do to the other side to maintain the balance and ensure the equation remains valid.

Think of an equation like a balanced scale. Both sides must weigh the same. If you add or subtract something from one side, you need to do the same on the other to keep it balanced. Similarly, if you multiply or divide one side by a number, you must do the same to the other side. This principle is the foundation of solving algebraic equations. Now that we understand the basic idea, let's move on to the first step: expanding the parentheses.

Step 1: Expanding the Parentheses

The first step in simplifying our equation is to get rid of the parentheses. We do this by using the distributive property, which states that a(b + c) = ab + ac. Let’s apply this to both sides of the equation:

• Left Side:

  • $5(t-1) = 5 * t - 5 * 1 = 5t - 5$
  • $3(t-3) = 3 * t - 3 * 3 = 3t - 9$

• Right Side:

  • $-2(t-1) = -2 * t - 2 * (-1) = -2t + 2$
  • $11(t+3) = 11 * t + 11 * 3 = 11t + 33$

Now, let’s substitute these expanded terms back into our equation. We have:

$5t - 5 + 3t - 9 = -2t + 2 + 11t + 33$

See how we've transformed the equation by simply applying the distributive property? This step is crucial because it allows us to combine like terms in the next step. It's like decluttering a room – by expanding the parentheses, we've made the equation less cluttered and easier to work with. Now, let's move on to the next step: combining those like terms!

Step 2: Combining Like Terms

Alright, now that we've expanded the parentheses, it's time to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (or no variable at all, which are called constants). In our equation, we have terms with 't' and constant terms. Let's group them together:

• Left Side:

  • Terms with 't': $5t + 3t = 8t$
  • Constant terms: $-5 - 9 = -14$

• Right Side:

  • Terms with 't': $-2t + 11t = 9t$
  • Constant terms: $2 + 33 = 35$

Now we can rewrite the equation with the combined terms:

$8t - 14 = 9t + 35$

Isn't it looking simpler already? Combining like terms is like organizing your tools before starting a project – it makes the whole process smoother and more efficient. We've reduced the number of terms in our equation, making it easier to isolate 't'. Speaking of isolating 't', that's exactly what we'll do in the next step. Let's move on to isolating the variable!

Step 3: Isolating the Variable

Okay, guys, this is where we start getting 't' all by itself! To isolate the variable 't', we need to get all the 't' terms on one side of the equation and all the constant terms on the other side. Let's start by moving the 't' terms to one side. A common approach is to subtract the smaller 't' term from both sides. In our equation, we have $8t$ on the left and $9t$ on the right, so let's subtract $8t$ from both sides:

$8t - 14 - 8t = 9t + 35 - 8t$

This simplifies to:

$-14 = t + 35$

Now we have 't' on the right side, but it's not quite alone yet because of the $+35$. To get rid of the $+35$, we need to subtract 35 from both sides:

$-14 - 35 = t + 35 - 35$

This gives us:

$-49 = t$

Ta-da! We've successfully isolated 't'! It's like solving a puzzle – each step brings us closer to the final piece. Now that we've found the value of 't', let's write out the final solution and make sure it matches one of the options given.

Step 4: The Solution

So, after all that work, we've found that $t = -49$. Let's check if this matches any of the options provided:

A. $t=-51$ B. $t=-53$ C. $t=-49$ D. $t=-14$ E. $t=-48$

It matches option C! Therefore, the correct answer is:

C. $t = -49$

Woohoo! We did it! Solving equations is like embarking on a journey, and we've reached our destination. But it's always a good idea to double-check our work, especially in math. Let's take a moment to verify our solution by substituting $t = -49$ back into the original equation.

Verifying the Solution

To verify our solution, we'll substitute $t = -49$ into the original equation: $5(t-1)+3(t-3)=-2(t-1)+11(t+3)$ and see if both sides of the equation are equal.

• Left Side:

  • $5(-49-1) + 3(-49-3)$
  • $5(-50) + 3(-52)$
  • $-250 - 156 = -406$

• Right Side:

  • $-2(-49-1) + 11(-49+3)$
  • $-2(-50) + 11(-46)$
  • $100 - 506 = -406$

Since both sides equal $-406$, our solution $t = -49$ is correct! It's always satisfying to confirm our answer and know we've done the math right. Verifying the solution is like proofreading your work before submitting it – it catches any potential errors and gives you confidence in your answer. Now that we've verified our solution, let's recap the steps we took to solve this equation.

Recap: Steps to Solve the Equation

Let's quickly recap the steps we took to solve the equation $5(t-1)+3(t-3)=-2(t-1)+11(t+3)$:

1. Expanding the Parentheses: We used the distributive property to remove the parentheses from the equation. This made it easier to combine like terms in the next step.
2. Combining Like Terms: We grouped together terms with the same variable ('t') and constant terms on each side of the equation. This simplified the equation and reduced the number of terms.
3. Isolating the Variable: We moved all the 't' terms to one side of the equation and all the constant terms to the other side. This allowed us to isolate 't' and find its value.
4. The Solution: We found that $t = -49$ is the solution to the equation.
5. Verifying the Solution: We substituted $t = -49$ back into the original equation to make sure both sides were equal. This confirmed that our solution was correct.

By following these steps, you can tackle many algebraic equations! Remember, practice makes perfect, so keep solving equations and you'll become a pro in no time. The key is to break down the problem into smaller, manageable steps and approach each step with confidence. Solving equations is like building a house – each step is a foundation for the next, and the final result is a strong, well-built solution.

Conclusion

So, guys, we've successfully solved the equation $5(t-1)+3(t-3)=-2(t-1)+11(t+3)$ and found that $t = -49$. We walked through each step in detail, from expanding the parentheses to verifying our solution. Remember, solving equations is a fundamental skill in mathematics, and by mastering these steps, you'll be well-equipped to tackle more complex problems. Keep practicing, and don't be afraid to ask for help when you need it. Math can be challenging, but it's also incredibly rewarding when you understand it. Keep up the great work, and happy solving!