Solving For T In The Equation (t+p)/q - Q/p = (t-q)/p + P/q

Introduction

In this article, we will explore the process of solving for the variables $t$, $x$, and $y$ in a given equation. The equation we will be working with is:

$\frac{t+p}{q}-\frac{q}{p}=\frac{t-q}{p}+\frac{p}{q}$

We aim to find the value of $t$ in terms of $p$ and $q$, under the condition that $p \neq q$. This problem falls under the category of algebraic equations, specifically dealing with fractions and variable manipulation. Solving such equations requires a systematic approach, including finding common denominators, simplifying expressions, and isolating the variable of interest. Let’s delve into the step-by-step solution to understand the underlying principles and techniques involved in solving this equation.

Problem Statement

We are given the equation:

$\frac{t+p}{q}-\frac{q}{p}=\frac{t-q}{p}+\frac{p}{q}$

Our objective is to solve for $t$, assuming that $p \neq q$. This condition is crucial because it prevents division by zero, which would make the equation undefined. The process of solving this equation involves algebraic manipulation to isolate $t$ on one side of the equation. We will begin by eliminating the fractions to simplify the equation and then proceed to collect like terms and solve for $t$. Understanding each step is essential for grasping the broader concepts of solving algebraic equations, which are fundamental in various fields of mathematics and science.

Step-by-Step Solution

1. Eliminate Fractions

To begin, we need to eliminate the fractions in the equation. This can be done by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is $pq$. Multiplying both sides by $pq$ gives us:

$pq(\frac{t+p}{q}-\frac{q}{p}) = pq(\frac{t-q}{p}+\frac{p}{q})$

Distributing $pq$ on both sides, we get:

$p(t+p) - q^2 = q(t-q) + p^2$

This step is crucial because it transforms the equation from one involving fractions to a simpler form involving only polynomials. Eliminating fractions often makes algebraic manipulation easier and reduces the chances of errors.

2. Expand and Simplify

Next, we expand the terms on both sides of the equation:

$pt + p^2 - q^2 = qt - q^2 + p^2$

Now, we can observe that $p^2$ and $-q^2$ appear on both sides of the equation, so we can cancel them out:

$pt = qt$

This simplification is a significant step as it reduces the complexity of the equation, making it easier to isolate the variable $t$. Recognizing and canceling out terms is a common technique in algebra that helps streamline the solution process.

3. Isolate t

To solve for $t$, we rearrange the equation to isolate $t$ on one side. We can rewrite the equation as:

$pt - qt = 0$

Factoring out $t$, we have:

$t(p - q) = 0$

Now, we divide both sides by $(p - q)$, but we must remember the condition that $p \neq q$, which ensures that we are not dividing by zero:

$t = \frac{0}{p - q}$

Since any non-zero number divided into zero is zero, we get:

$t = 0$

Therefore, the solution for $t$ is 0, provided that $p \neq q$. This step highlights the importance of considering the conditions given in the problem statement to avoid invalid operations.

Final Answer

The solution for $t$ in the equation $\frac{t+p}{q}-\frac{q}{p}=\frac{t-q}{p}+\frac{p}{q}$, given that $p \neq q$, is:

$t = 0$

This result indicates that the value of $t$ that satisfies the given equation is zero, under the condition that $p$ is not equal to $q$. This solution is obtained through careful algebraic manipulation, including eliminating fractions, simplifying terms, and isolating the variable $t$. The process demonstrates the fundamental principles of solving algebraic equations, which are applicable in various mathematical and scientific contexts.

Verification

To verify our solution, we can substitute $t = 0$ back into the original equation:

$\frac{0+p}{q}-\frac{q}{p}=\frac{0-q}{p}+\frac{p}{q}$

This simplifies to:

$\frac{p}{q}-\frac{q}{p}=-\frac{q}{p}+\frac{p}{q}$

Rearranging the terms, we get:

$\frac{p}{q}-\frac{q}{p}=\frac{p}{q}-\frac{q}{p}$

Since both sides of the equation are equal, our solution $t = 0$ is verified. This verification step is crucial to ensure that the solution obtained is correct and satisfies the original equation. It provides confidence in the algebraic manipulations performed and confirms the accuracy of the result.

Conclusion

In conclusion, we have successfully solved for $t$ in the equation $\frac{t+p}{q}-\frac{q}{p}=\frac{t-q}{p}+\frac{p}{q}$, given the condition that $p \neq q$. The solution we found is $t = 0$. This was achieved through a series of algebraic manipulations, including eliminating fractions, expanding terms, simplifying the equation, and isolating the variable $t$. The process involved careful application of algebraic principles and a systematic approach to problem-solving.

Key Steps Summary:

1. Eliminated fractions by multiplying both sides of the equation by the least common multiple of the denominators.
2. Expanded and simplified the equation by distributing terms and canceling out like terms.
3. Isolated t by rearranging the equation and factoring out $t$.
4. Verified the solution by substituting $t = 0$ back into the original equation.

The verification step confirmed the accuracy of our solution, demonstrating the importance of checking results in mathematical problem-solving. This exercise highlights the fundamental principles of algebra and the techniques used to solve equations, which are essential skills in mathematics and various related fields. The step-by-step approach used in this solution can be applied to a wide range of algebraic problems, providing a solid foundation for more advanced mathematical concepts.