Given: $y = \frac{14mn}{m+n}$ And $p:q = Q:r$. Show That: $\frac{p}{r} = \frac{p^2 + Q^2}{q^2 + R^2}$. Prove That: $p^4q^4r^4(\frac{1}{p^6} + \frac{1}{q^6} + \frac{1}{r^6}) = P^6 + Q^6 + R^6$. Find The Value Of: $\frac{y+7m}{y-7m} + \frac{y+7n}{y-7n}$.

This article delves into several intriguing mathematical problems, offering detailed explanations and step-by-step solutions. We will explore the properties of proportions, algebraic manipulations, and the application of these concepts to solve complex equations. The problems we will address include proving relationships between ratios, demonstrating algebraic identities, and evaluating expressions involving fractions. Understanding these concepts is crucial for anyone looking to enhance their mathematical skills and tackle advanced problems.

Problem 1: Proving Proportional Relationships

We are given that $y = \frac{14mn}{m+n}$ and $p:q = q:r$. Our first task is to show that $\frac{p}{r} = \frac{p^2 + q^2}{q^2 + r^2}$. This problem involves manipulating ratios and proportions to arrive at the desired conclusion. The given condition $p:q = q:r$ tells us that $q$ is the geometric mean of $p$ and $r$. This relationship is key to unlocking the solution.

Step-by-Step Solution

To begin, let’s express the given proportion $p:q = q:r$ as an equation. This can be written as $\frac{p}{q} = \frac{q}{r}$. From this equation, we can derive a crucial relationship: $q^2 = pr$. This equation will be instrumental in simplifying our expressions and proving the required equality. Now, we aim to show that $\frac{p}{r} = \frac{p^2 + q^2}{q^2 + r^2}$. Let’s start by substituting $q^2$ with $pr$ in the numerator and the denominator of the right-hand side of the equation. This substitution allows us to express the entire equation in terms of $p$ and $r$, making it easier to manipulate.

Substituting $q^2 = pr$ into the right-hand side, we get:

$$\frac{p^2 + q^2}{q^2 + r^2} = \frac{p^2 + pr}{pr + r^2}$$

Now, we can factor out common terms from both the numerator and the denominator. In the numerator, we can factor out $p$, and in the denominator, we can factor out $r$. This factoring step is essential for simplifying the fraction and revealing the underlying relationship between $p$ and $r$.

Factoring out common terms, we have:

$$\frac{p^2 + pr}{pr + r^2} = \frac{p(p + r)}{r(p + r)}$$

We now observe that $(p + r)$ is a common factor in both the numerator and the denominator. We can cancel out this common factor, provided that $p + r \neq 0$. This step simplifies the fraction significantly and brings us closer to the desired result.

Canceling the common factor $(p + r)$, we obtain:

$$\frac{p(p + r)}{r(p + r)} = \frac{p}{r}$$

Thus, we have successfully shown that $\frac{p^2 + q^2}{q^2 + r^2} = \frac{p}{r}$, given that $p:q = q:r$. This proof demonstrates the power of algebraic manipulation and the use of proportions to derive complex relationships.

Conclusion

This first problem highlights the importance of understanding proportional relationships and how they can be manipulated algebraically to prove equalities. By recognizing that $q^2 = pr$ and substituting this into the expression, we were able to simplify the fraction and arrive at the desired result. This type of problem is fundamental in many areas of mathematics and demonstrates the elegance of mathematical proofs.

Problem 2: Proving an Algebraic Identity

Our next challenge is to prove that $p^4q^4r^4(\frac{1}{p^6} + \frac{1}{q^6} + \frac{1}{r^6}) = p^6 + q^6 + r^6$, given the same condition $p:q = q:r$. This problem involves algebraic manipulation and a deep understanding of exponents and fractions. The key to solving this problem lies in simplifying the left-hand side of the equation and showing that it is equivalent to the right-hand side.

Step-by-Step Solution

To begin, let’s rewrite the left-hand side of the equation by distributing $p^4q^4r^4$ across the terms inside the parentheses. This step is crucial for simplifying the expression and revealing potential cancellations and simplifications.

Distributing $p^4q^4r^4$, we get:

$$p^4q^4r^4(\frac{1}{p^6} + \frac{1}{q^6} + \frac{1}{r^6}) = \frac{p^4q^4r^4}{p^6} + \frac{p^4q^4r^4}{q^6} + \frac{p^4q^4r^4}{r^6}$$

Now, we can simplify each term by canceling out common factors in the numerators and denominators. This simplification step is essential for reducing the complexity of the expression and making it easier to work with.

Simplifying each term, we have:

$$\frac{p^4q^4r^4}{p^6} + \frac{p^4q^4r^4}{q^6} + \frac{p^4q^4r^4}{r^6} = \frac{q^4r^4}{p^2} + \frac{p^4r^4}{q^2} + \frac{p^4q^4}{r^2}$$

Recall that from the given condition $p:q = q:r$, we have $q^2 = pr$. We can use this relationship to further simplify the expression. Substituting $q^2 = pr$ into the terms, we can express the entire equation in terms of $p$ and $r$, which will help us in reaching the final result.

Substituting $q^2 = pr$, we get:

$$\frac{q^4r^4}{p^2} + \frac{p^4r^4}{q^2} + \frac{p^4q^4}{r^2} = \frac{(pr)^2r^4}{p^2} + \frac{p^4r^4}{pr} + \frac{p^4(pr)^2}{r^2}$$

Further simplifying, we have:

$$\frac{p^2r^6}{p^2} + p^3r^3 + \frac{p^6r^2}{r^2} = r^6 + p^3r^3 + p^6$$

Now, let's look at the right-hand side of the original equation, which is $p^6 + q^6 + r^6$. We need to express $q^6$ in terms of $p$ and $r$ using the relationship $q^2 = pr$.

We have $q^6 = (q^2)^3 = (pr)^3 = p^3r^3$. Substituting this into the right-hand side, we get:

$$p^6 + q^6 + r^6 = p^6 + p^3r^3 + r^6$$

Comparing this with the simplified left-hand side, we can see that both sides are equal:

$$r^6 + p^3r^3 + p^6 = p^6 + p^3r^3 + r^6$$

Thus, we have successfully proven that $p^4q^4r^4(\frac{1}{p^6} + \frac{1}{q^6} + \frac{1}{r^6}) = p^6 + q^6 + r^6$.

Conclusion

This problem demonstrates the importance of algebraic manipulation, substitution, and simplification in proving complex identities. By carefully distributing terms, canceling common factors, and using the given relationship $q^2 = pr$, we were able to show the equality of both sides of the equation. This problem reinforces the idea that complex algebraic expressions can be simplified through systematic manipulation and a solid understanding of algebraic principles.

Problem 3: Evaluating an Expression

Finally, we are given $y = \frac{14mn}{m+n}$ and asked to find the value of $\frac{y+7m}{y-7m} + \frac{y+7n}{y-7n}$. This problem involves substituting the given value of $y$ into the expression and simplifying it to find a numerical value. The key to solving this problem is to perform the substitutions carefully and use algebraic techniques to combine and simplify the fractions.

Step-by-Step Solution

First, let’s substitute the given value of $y = \frac{14mn}{m+n}$ into the expression $\frac{y+7m}{y-7m} + \frac{y+7n}{y-7n}$. This substitution is the first step in simplifying the expression and will allow us to work with the terms more effectively.

Substituting $y$, we get:

$$\frac{\frac{14mn}{m+n}+7m}{\frac{14mn}{m+n}-7m} + \frac{\frac{14mn}{m+n}+7n}{\frac{14mn}{m+n}-7n}$$

Next, we need to simplify the fractions within the larger expression. To do this, we will find a common denominator for the terms in the numerators and denominators of the fractions. The common denominator for each fraction will be $(m+n)$. This step is crucial for combining the terms and simplifying the expression.

Simplifying the fractions, we have:

$$\frac{\frac{14mn + 7m(m+n)}{m+n}}{\frac{14mn - 7m(m+n)}{m+n}} + \frac{\frac{14mn + 7n(m+n)}{m+n}}{\frac{14mn - 7n(m+n)}{m+n}}$$

Now, we can cancel out the common denominator $(m+n)$ in each fraction. This cancellation simplifies the expression and makes it easier to work with.

Canceling the common denominator, we get:

$$\frac{14mn + 7m(m+n)}{14mn - 7m(m+n)} + \frac{14mn + 7n(m+n)}{14mn - 7n(m+n)}$$

Next, we need to expand the terms in the numerators and denominators. Expanding the terms will help us to combine like terms and further simplify the expression.

Expanding the terms, we have:

$$\frac{14mn + 7m^2 + 7mn}{14mn - 7m^2 - 7mn} + \frac{14mn + 7nm + 7n^2}{14mn - 7nm - 7n^2}$$

Now, we can combine like terms in the numerators and denominators. This step simplifies the expression further and prepares it for the next step of simplification.

Combining like terms, we get:

$$\frac{21mn + 7m^2}{7mn - 7m^2} + \frac{21mn + 7n^2}{7mn - 7n^2}$$

We can factor out common factors from the numerators and denominators. In the first fraction, we can factor out $7m$, and in the second fraction, we can factor out $7n$. This factoring step is essential for simplifying the fractions and revealing the underlying structure of the expression.

Factoring out common factors, we have:

$$\frac{7m(3n + m)}{7m(n - m)} + \frac{7n(3m + n)}{7n(m - n)}$$

Now, we can cancel the common factors $7m$ and $7n$ in the fractions. This cancellation simplifies the fractions significantly and brings us closer to the final result.

Canceling the common factors, we get:

$$\frac{3n + m}{n - m} + \frac{3m + n}{m - n}$$

To add these fractions, we need a common denominator. The common denominator is $(n - m)$. We can rewrite the second fraction with the denominator $(n - m)$ by multiplying the numerator and denominator by $-1$.

Rewriting the second fraction, we have:

$$\frac{3n + m}{n - m} - \frac{3m + n}{n - m}$$

Now, we can combine the fractions by adding the numerators over the common denominator.

Combining the fractions, we get:

$$\frac{(3n + m) - (3m + n)}{n - m}$$

Simplifying the numerator, we have:

$$\frac{3n + m - 3m - n}{n - m} = \frac{2n - 2m}{n - m}$$

Finally, we can factor out a $2$ from the numerator and simplify the fraction.

Factoring and simplifying, we get:

$$\frac{2(n - m)}{n - m} = 2$$

Thus, the value of the expression $\frac{y+7m}{y-7m} + \frac{y+7n}{y-7n}$ is $2$.

Conclusion

This problem demonstrates the importance of careful substitution, simplification, and algebraic manipulation in evaluating expressions. By substituting the given value of $y$, finding common denominators, and factoring out common terms, we were able to simplify the expression and arrive at a numerical value. This problem reinforces the idea that complex expressions can be evaluated through a systematic approach and a solid understanding of algebraic principles.

In this article, we have explored three distinct mathematical problems, each requiring a unique set of skills and techniques. We have demonstrated how to prove proportional relationships, algebraic identities, and evaluate complex expressions. These problems highlight the importance of understanding algebraic principles, mastering algebraic manipulation, and approaching problems systematically. By working through these examples, readers can enhance their mathematical problem-solving skills and gain a deeper appreciation for the elegance and power of mathematics.

Keywords

• Proportional Relationships: Understanding proportions and how to manipulate them algebraically is crucial for solving a variety of mathematical problems.
• Algebraic Identities: Proving algebraic identities involves the strategic use of substitutions, factoring, and simplification techniques.
• Evaluating Expressions: Evaluating complex expressions requires careful substitution, simplification, and a solid understanding of algebraic principles.
• Mathematical Proofs: Constructing mathematical proofs involves a logical sequence of steps that lead to a desired conclusion.
• Fraction Simplification: Simplifying fractions is a fundamental skill in algebra and is essential for solving equations and evaluating expressions.