Solving For M In The Equation (p/(m+n)) = (q/(m-n)) + (3/(m+n))

This article delves into the intricate process of isolating the variable 'm' in a given algebraic equation. We will methodically dissect the equation ${\frac{p}{m+n} = \frac{q}{m-n} + \frac{3}{m+n}}$, employing algebraic manipulations to express 'm' explicitly in terms of the other variables: 'p', 'q', and 'n'. This comprehensive guide aims to provide a clear, step-by-step solution, enhancing your understanding of algebraic problem-solving techniques. Understanding how to solve for a specific variable within a complex equation is a fundamental skill in algebra and has wide-ranging applications in various fields, including physics, engineering, and economics. The ability to manipulate equations and isolate variables allows us to model real-world scenarios, make predictions, and gain valuable insights. In this article, we will not only focus on the mechanics of solving for 'm' but also emphasize the underlying principles and strategies that can be applied to a broader range of algebraic problems. By the end of this guide, you will be equipped with the knowledge and skills necessary to tackle similar challenges with confidence.

1. Initial Equation and Objective

We begin with the equation ${\frac{p}{m+n} = \frac{q}{m-n} + \frac{3}{m+n}}$, where our primary objective is to isolate 'm' on one side of the equation. This involves a series of algebraic manipulations, including combining fractions, cross-multiplication, and rearrangement of terms. The initial equation presents a common scenario in algebra where the variable we want to solve for is present in multiple terms and within denominators. This requires a strategic approach to simplify the equation and gradually isolate 'm'. Our first step will involve combining like terms, specifically the fractions with the denominator ${m+n}$. This will help us consolidate the equation and pave the way for further simplification. By carefully applying algebraic principles, we will navigate through each step, ensuring that we maintain the equality of the equation while progressing towards our goal of expressing 'm' in terms of 'p', 'q', and 'n'.

2. Combining Like Terms

In this equation, the terms ${\frac{p}{m+n}}$ and ${\frac{3}{m+n}}$ share a common denominator. We can simplify the equation by combining these terms. This is a crucial step in isolating 'm' because it reduces the number of fractions we need to deal with. Combining like terms is a fundamental algebraic technique that simplifies equations and makes them easier to solve. By adding or subtracting terms with the same denominator, we can consolidate the equation and reduce its complexity. In this case, combining the terms with the denominator ${m+n}$ will allow us to eliminate one of the fractions, making the equation more manageable. This step highlights the importance of recognizing and utilizing opportunities for simplification in algebraic problem-solving. By carefully examining the equation and identifying like terms, we can streamline the solution process and avoid unnecessary complications. The ability to combine like terms is a cornerstone of algebraic manipulation and a skill that is essential for solving a wide range of equations.

Detailed Steps for Combining Like Terms

To combine the like terms, we add ${\frac{3}{m+n}}$ to ${\frac{p}{m+n}}$. This results in a single fraction with the same denominator: ${\frac{p+3}{m+n}}$. Now, the equation looks like this: ${\frac{p+3}{m+n} = \frac{q}{m-n}}$. This simplification is a significant step forward, as it reduces the number of fractions from three to two. By combining the like terms, we have effectively consolidated the equation and made it more amenable to further manipulation. This step demonstrates the power of algebraic simplification in making complex equations more tractable. The resulting equation, ${\frac{p+3}{m+n} = \frac{q}{m-n}}$, is now in a form that allows us to proceed with cross-multiplication, a technique that will help us eliminate the fractions altogether. This is a crucial step in our journey to isolate 'm' and express it in terms of the other variables.

3. Cross-Multiplication

Now, cross-multiply to eliminate the fractions. Cross-multiplication is a powerful technique used to solve equations involving fractions. It involves multiplying the numerator of one fraction by the denominator of the other and vice versa. This effectively eliminates the fractions, transforming the equation into a more manageable form. In this case, we will multiply ${(p+3)}$ by ${(m-n)}$ and ${q}$ by ${(m+n)}$. This step is crucial because it removes the fractions, allowing us to work with a simpler algebraic expression. Cross-multiplication is a valuable tool in algebraic manipulation, and understanding its application is essential for solving a wide range of equations. By applying this technique, we are moving closer to our goal of isolating 'm' and expressing it in terms of the other variables. The resulting equation will be a linear equation in 'm', which can be solved using standard algebraic methods.

Applying Cross-Multiplication

By cross-multiplying, we get ${(p+3)(m-n) = q(m+n)}$. This step transforms the equation from a fractional form to a more straightforward algebraic expression. The elimination of fractions simplifies the equation and makes it easier to manipulate. The resulting equation, ${(p+3)(m-n) = q(m+n)}$, is now in a form that allows us to proceed with expanding the products. This will involve applying the distributive property, a fundamental algebraic principle that allows us to multiply a sum or difference by a single term. By expanding the products, we will further simplify the equation and bring us closer to isolating 'm'. This step highlights the importance of algebraic manipulation in transforming equations into a form that is easier to solve.

4. Expanding the Products

Expand both sides of the equation to remove the parentheses. This involves applying the distributive property, a fundamental principle in algebra. Expanding the products is a crucial step in simplifying the equation and isolating 'm'. By removing the parentheses, we can combine like terms and rearrange the equation to solve for 'm'. The distributive property allows us to multiply a sum or difference by a single term, which is essential for expanding algebraic expressions. In this case, we will expand both the left and right sides of the equation, resulting in a more complex expression but one that is easier to manipulate. This step demonstrates the importance of algebraic manipulation in transforming equations into a form that is easier to solve. The expanded equation will allow us to group terms containing 'm' on one side and terms without 'm' on the other, bringing us closer to our goal of isolating 'm'.

Detailed Expansion

Expanding ${(p+3)(m-n)}$ gives us ${pm - pn + 3m - 3n}$, and expanding ${q(m+n)}$ yields ${qm + qn}$. Thus, the equation becomes ${pm - pn + 3m - 3n = qm + qn}$. This step is a critical transformation, as it eliminates the parentheses and allows us to rearrange the terms. The expanded equation now contains individual terms, some of which involve 'm' and others that do not. This separation of terms is essential for isolating 'm' on one side of the equation. By carefully applying the distributive property, we have created an equation that is more amenable to further manipulation. The next step will involve grouping the terms containing 'm' on one side and the remaining terms on the other, bringing us closer to our goal of expressing 'm' in terms of 'p', 'q', and 'n'.

5. Grouping Terms with 'm'

Collect all terms containing 'm' on one side of the equation and the remaining terms on the other side. This is a fundamental step in solving for a variable in an algebraic equation. By grouping the terms containing 'm' on one side, we can isolate 'm' and express it in terms of the other variables. This process involves adding or subtracting terms from both sides of the equation to move them to the desired location. The goal is to have all the terms with 'm' on one side and all the terms without 'm' on the other. This step is crucial for simplifying the equation and making it easier to solve for 'm'. Grouping terms is a common technique in algebra and is essential for solving a wide range of equations.

Isolating 'm' Terms

To group the terms with 'm', we subtract ${qm}$ from both sides and add ${pn}$ and ${3n}$ to both sides. This gives us ${pm + 3m - qm = qn + pn + 3n}$. This rearrangement is a significant step towards isolating 'm'. By moving all the terms containing 'm' to the left side and the remaining terms to the right side, we have created a structure that allows us to factor out 'm'. The resulting equation, ${pm + 3m - qm = qn + pn + 3n}$, is now in a form that is conducive to factoring. Factoring out 'm' will be the next step in our process of solving for 'm' and expressing it in terms of the other variables.

6. Factoring Out 'm'

Factor out 'm' from the terms on the left side of the equation. Factoring is a crucial algebraic technique that involves expressing a sum or difference as a product. In this case, we will factor out 'm' from the terms on the left side of the equation, which will allow us to isolate 'm' more effectively. Factoring is the reverse process of distribution and is a fundamental skill in algebra. By factoring out 'm', we will be able to express it as a coefficient multiplied by a single term, making it easier to solve for. This step is a key step in our process of isolating 'm' and expressing it in terms of the other variables.

Applying Factoring

Factoring out 'm' gives us ${m(p + 3 - q) = qn + pn + 3n}$. This step is a critical simplification, as it isolates 'm' as a single factor. The equation is now in a form where we can easily solve for 'm' by dividing both sides by the expression in parentheses. Factoring is a powerful technique in algebra, and its application here demonstrates its effectiveness in simplifying equations. The resulting equation, ${m(p + 3 - q) = qn + pn + 3n}$, is now just one step away from expressing 'm' in terms of 'p', 'q', and 'n'. The next step will involve dividing both sides by ${(p + 3 - q)}$, which will finally isolate 'm'.

7. Isolating 'm'

Divide both sides by ${(p + 3 - q)}$ to isolate 'm'. This is the final step in solving for 'm' in terms of the other variables. Division is the inverse operation of multiplication, and in this case, it will undo the multiplication by ${(p + 3 - q)}$ on the left side of the equation. By dividing both sides by this expression, we will isolate 'm' and express it explicitly in terms of 'p', 'q', and 'n'. This step is the culmination of all the previous algebraic manipulations and represents the solution to the problem. The resulting equation will provide a formula for calculating 'm' given the values of 'p', 'q', and 'n'.

The Final Solution

Dividing both sides by ${(p + 3 - q)}$, we get: ${ m = \frac{qn + pn + 3n}{p + 3 - q} }$. This equation expresses 'm' in terms of 'p', 'q', and 'n'. This is the final solution to the problem, and it provides a formula for calculating 'm' given the values of 'p', 'q', and 'n'. The solution demonstrates the power of algebraic manipulation in solving complex equations. By systematically applying algebraic principles, we were able to isolate 'm' and express it in terms of the other variables. This result can be used to solve for 'm' in various contexts, highlighting the practical applications of algebra.

8. Conclusion

In summary, we have successfully solved for 'm' in the equation ${\frac{p}{m+n} = \frac{q}{m-n} + \frac{3}{m+n}}$. The solution, ${ m = \frac{qn + pn + 3n}{p + 3 - q} }$, expresses 'm' in terms of 'p', 'q', and 'n'. This process involved combining like terms, cross-multiplication, expanding products, grouping terms, factoring, and finally, isolating 'm'. This step-by-step guide provides a clear and comprehensive approach to solving this type of algebraic problem. The techniques used in this solution can be applied to a wide range of algebraic equations, making this a valuable exercise in algebraic problem-solving. The ability to solve for a specific variable in an equation is a fundamental skill in mathematics and has numerous applications in various fields. By mastering these techniques, you can confidently tackle similar challenges and gain a deeper understanding of algebraic principles.

Key Takeaways

• Combining like terms is a crucial first step in simplifying equations.
• Cross-multiplication is effective for eliminating fractions.
• Expanding products allows for rearrangement and grouping of terms.
• Factoring is essential for isolating the desired variable.
• Systematic algebraic manipulation is key to solving complex equations.

By following these steps and understanding the underlying principles, you can confidently solve for any variable in a complex algebraic equation.