Solving For T: A Step-by-Step Guide

Nov 13, 2025 by Admin 36 views

Let's break down how to solve the equation $p = s_1t - s_2t$ for $t$ . This is a common type of algebraic problem, and understanding the steps will help you tackle similar equations with confidence. We'll go through the process in detail, explaining each step along the way so you can see how we arrive at the correct answer. By the end of this guide, you'll not only know the solution but also understand the reasoning behind it. So, let's dive in and get started!

Understanding the Problem

The given equation is $p = s_1t - s_2t$ . Our goal is to isolate $t$ on one side of the equation. This means we want to rewrite the equation in the form $t = \{some expression\}$ . To do this, we'll use algebraic manipulation, which involves performing the same operations on both sides of the equation to maintain balance.

The key to solving this equation lies in recognizing that $t$ is a common factor in the terms $s_1t$ and $-s_2t$ . By factoring out $t$ , we can simplify the equation and make it easier to isolate $t$ . This is a standard technique in algebra, and it's something you'll use frequently when solving for variables.

Think of it like this: if you have $5x - 3x$ , you can combine those terms to get $2x$ because $x$ is a common factor. Similarly, in our equation, $t$ is the common factor that we can factor out. Understanding this concept is crucial for successfully solving the equation.

Now, let's move on to the next step, where we'll actually factor out the $t$ and simplify the equation.

Step-by-Step Solution

Factor out t:

We can factor out $t$ from the right side of the equation: $p = t(s_1 - s_2)$

This step is the heart of the solution. By factoring out $t$ , we've transformed the equation from having two terms with $t$ to a single term with $t$ . This makes it much easier to isolate $t$ . Remember, factoring is the reverse of distribution. If we were to distribute $t$ back into the parentheses, we would get back our original equation: $t * s_1 - t * s_2 = s_1t - s_2t$ .

Factoring is a powerful tool in algebra, and it's essential to master it. It allows you to simplify complex expressions and solve equations that would otherwise be difficult to manage. Practice factoring with different types of expressions to become more comfortable with this technique.

Isolate t:

To isolate $t$ , we need to divide both sides of the equation by $(s_1 - s_2)$ : $\frac{p}{s_1 - s_2} = \frac{t(s_1 - s_2)}{s_1 - s_2}$

This step is based on the principle that you can perform the same operation on both sides of an equation without changing its balance. Dividing both sides by $(s_1 - s_2)$ cancels out the $(s_1 - s_2)$ on the right side, leaving us with $t$ isolated.

It's important to note that we're assuming that $(s_1 - s_2)$ is not equal to zero. If $(s_1 - s_2)$ were equal to zero, then we would be dividing by zero, which is undefined. In such a case, the equation would have either no solution or infinitely many solutions, depending on the value of $p$ .

Simplify:

The $(s_1 - s_2)$ terms on the right side cancel out, leaving us with: $t = \frac{p}{s_1 - s_2}$

This is the final solution. We have successfully isolated $t$ and expressed it in terms of $p$ , $s_1$ , and $s_2$ .

The solution tells us that the value of $t$ is equal to $p$ divided by the difference between $s_1$ and $s_2$ . This relationship can be useful in various applications, depending on what $p$ , $s_1$ , and $s_2$ represent.

Analyzing the Options

Now, let's compare our solution with the given options:

A. $t=p(s_1-s_2)$ B. $t=1-p-s_1+s_2$ C. $t=\frac{p}{s_1-s_2}$ D. $t=\frac{P}{S_1+S_2}$

Option C, $t=\frac{p}{s_1-s_2}$ , matches our solution exactly. Therefore, option C is the correct answer.

Options A, B, and D are incorrect. Option A is incorrect because it multiplies $p$ by $(s_1 - s_2)$ instead of dividing. Option B is incorrect because it involves subtracting and adding terms in a way that doesn't follow from the original equation. Option D is incorrect because it divides $p$ by $(S_1 + S_2)$ instead of $(s_1 - s_2)$ .

By carefully comparing our solution with the given options, we can confidently identify the correct answer and avoid making mistakes.

Common Mistakes to Avoid

Forgetting to factor: A common mistake is trying to isolate $t$ without first factoring it out. This makes the problem much more difficult, if not impossible, to solve.
Incorrectly dividing: When dividing both sides of the equation, make sure to divide the entire side by the term you're dividing by. Don't just divide individual terms.
Not considering the case where $s_1 - s_2 = 0$ : While not explicitly required in this problem, it's good practice to consider cases where the denominator might be zero. This can lead to undefined solutions.
Algebra Errors: Ensure each step you take algebraically is sound and follows the rules of algebra. A small mistake will lead to the wrong answer.

Algebra can be tough, you guys! But by avoiding these mistakes and practicing regularly, you can improve your skills and solve equations more accurately and efficiently. Remember to double-check your work and pay attention to detail.

Conclusion

The correct equation solved for $t$ is $t = \frac{p}{s_1 - s_2}$ . We arrived at this solution by factoring out $t$ from the original equation and then dividing both sides by $(s_1 - s_2)$ to isolate $t$ . Understanding the steps involved and avoiding common mistakes will help you solve similar algebraic problems with ease. Keep practicing, and you'll become a pro at solving for variables in no time! Remember to always double-check your work and think critically about each step you take. With a little bit of effort, you can master algebra and unlock a whole new world of mathematical possibilities.