Solving 1/m + 1/n + 1/x = 1/(m + N + X) Find X

The equation ${ \frac{1}{m} + \frac{1}{n} + \frac{1}{x} = \frac{1}{m+n+x} }$ where ${ m + n + x \neq 0 }$ presents an intriguing mathematical problem. This article delves into a step-by-step solution to this equation, exploring the underlying concepts and providing a clear, comprehensive explanation suitable for anyone interested in mathematical problem-solving. Our exploration will not only focus on arriving at the correct solutions but also on understanding the logic and reasoning behind each step. We will dissect the equation, manipulate it algebraically, and ultimately reveal the possible values of x that satisfy the given condition. This journey through the equation will enhance your understanding of algebraic manipulations, equation-solving techniques, and the beauty of mathematical problem-solving.

Understanding the Problem

Before we dive into the solution, let's first understand the problem. The equation ${ \frac{1}{m} + \frac{1}{n} + \frac{1}{x} = \frac{1}{m+n+x} }$ involves reciprocals of variables and a condition that ${ m + n + x \neq 0 }$. This condition is crucial because division by zero is undefined in mathematics. The goal is to find the values of x that make the equation true, given m and n are constants. We need to manipulate this equation in such a way that we can isolate x and determine its possible values. The complexity arises from the fractions and the presence of x in both the numerator and the denominator. Therefore, a strategic approach involving algebraic manipulation is essential to unravel the solution. Our initial steps will involve combining fractions and simplifying the equation to a more manageable form.

Step-by-Step Solution

To solve the equation, we will follow a series of algebraic manipulations. First, we'll combine the fractions on the left-hand side of the equation. This involves finding a common denominator and adding the numerators. Then, we'll cross-multiply to eliminate the fractions, which will lead to a polynomial equation. Finally, we'll solve this polynomial equation to find the values of x. This step-by-step approach ensures clarity and allows us to break down the problem into smaller, more manageable parts. Each step is crucial and builds upon the previous one, ultimately leading us to the solution. By meticulously following these steps, we can systematically solve the equation and gain a deeper understanding of the underlying mathematical principles.

1. Combine Fractions on the Left-Hand Side

We begin by combining the fractions ${ \frac{1}{m} }$, ${ \frac{1}{n} }$, and ${ \frac{1}{x} }$ on the left-hand side of the equation. To do this, we need to find a common denominator, which in this case is mnx. So, we rewrite the fractions with the common denominator:

${ \frac{1}{m} + \frac{1}{n} + \frac{1}{x} = \frac{nx}{mnx} + \frac{mx}{mnx} + \frac{mn}{mnx} }$

Now, we can add the numerators:

${ \frac{nx + mx + mn}{mnx} }$

So, our equation now becomes:

${ \frac{nx + mx + mn}{mnx} = \frac{1}{m+n+x} }$

This step is critical as it consolidates the left-hand side into a single fraction, making the equation easier to handle. The common denominator mnx allows us to combine the terms effectively and prepare for the next step, which involves eliminating the fractions altogether.

2. Cross-Multiplication

To eliminate the fractions, we perform cross-multiplication. This involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. Cross-multiplication is a valid algebraic operation as long as we ensure that the denominators are not zero, which is already guaranteed by the condition ${ m + n + x \neq 0 }$ and the implicit assumption that ${ m, n, x \neq 0 }$. Performing the cross-multiplication, we get:

${ (nx + mx + mn)(m + n + x) = mnx(1) }$

Expanding this product will give us a polynomial equation in terms of x. This step is a significant simplification, as it transforms the fractional equation into a more manageable polynomial equation. The resulting polynomial equation will be the key to unlocking the solutions for x.

3. Expand and Simplify

Now, let's expand the left-hand side of the equation:

${ (nx + mx + mn)(m + n + x) = nx(m + n + x) + mx(m + n + x) + mn(m + n + x) }$

Expanding each term, we get:

${ nmx + n^2x + nx^2 + m^2x + mnx + mx^2 + m^2n + mn^2 + mnx }$

Combining like terms, the equation becomes:

${ nx^2 + mx^2 + nmx + n^2x + m^2x + mnx + mnx + m^2n + mn^2 = mnx }$

Simplifying further, we have:

${ nx^2 + mx^2 + n^2x + m^2x + 2mnx + m^2n + mn^2 = 0 }$

This simplified polynomial equation is now in a form that we can factorize and solve for x. This expansion and simplification are crucial steps in solving the equation, as they transform the complex expression into a more structured form, making it easier to identify potential solutions.

4. Rearrange and Factor

Rearrange the terms to group them in a way that facilitates factorization:

${ nx^2 + mx^2 + n^2x + m^2x + 2mnx + m^2n + mn^2 = mnx }$

${ x^2(n + m) + x(n^2 + m^2 + 2mn) + mn(m + n) = 0 }$

Notice that ${ n^2 + m^2 + 2mn }$ can be factored as ${ (m + n)^2 }$, so the equation becomes:

${ x^2(m + n) + x(m + n)^2 + mn(m + n) = 0 }$

Now, we can factor out ${ (m + n) }$ from the entire equation:

${ (m + n)(x^2 + x(m + n) + mn) = 0 }$

Since ${ m + n + x \neq 0 }$, we consider the case where ${ m + n \neq 0 }$. Thus, we can divide both sides by ${ (m + n) }$:

${ x^2 + x(m + n) + mn = 0 }$

This quadratic equation is now much simpler to solve. Factoring out ${ (m + n) }$ was a key step in simplifying the equation and making it solvable. The next step involves factoring the quadratic equation itself to find the roots, which represent the possible values of x.

5. Solve the Quadratic Equation

Now, we have the quadratic equation:

${ x^2 + x(m + n) + mn = 0 }$

We can factor this quadratic equation as:

${ (x + m)(x + n) = 0 }$

This gives us two possible solutions for x:

${ x + m = 0 \quad \text{or} \quad x + n = 0 }$

Solving for x in each case, we get:

${ x = -m \quad \text{or} \quad x = -n }$

These are the solutions to the equation. By factoring the quadratic equation, we were able to easily identify the values of x that satisfy the original equation. This step completes the solution process, providing us with the possible values of x.

Final Answer

The solutions to the equation ${ \frac{1}{m} + \frac{1}{n} + \frac{1}{x} = \frac{1}{m+n+x} }$ are:

${ x = -m \quad \text{or} \quad x = -n }$

Therefore, the correct option is (b) x = -m; x = -n. This comprehensive solution demonstrates the power of algebraic manipulation and problem-solving techniques in unraveling complex equations. By meticulously following each step, we have arrived at the final answer, showcasing the elegance and precision of mathematical reasoning.

Conclusion

In this article, we successfully solved the equation ${ \frac{1}{m} + \frac{1}{n} + \frac{1}{x} = \frac{1}{m+n+x} }$ by employing a systematic approach involving algebraic manipulation, simplification, and factorization. We started by combining fractions, then cross-multiplied to eliminate the fractions, expanded and simplified the resulting polynomial, factored the equation to isolate x, and finally, solved for x. The solutions we found were ${ x = -m }$ and ${ x = -n }$. This exercise highlights the importance of understanding algebraic techniques and the power of strategic problem-solving in mathematics. By breaking down the problem into smaller, manageable steps, we were able to navigate the complexities of the equation and arrive at the correct solutions. This process not only provides the answer but also enhances our understanding of mathematical principles and problem-solving strategies.

This detailed solution provides a clear and comprehensive guide to solving this type of equation, emphasizing the logical steps and reasoning involved. Understanding these steps is crucial for tackling similar problems and for developing a strong foundation in algebra and equation-solving.