- Find a common denominator for the numerator (\(\frac{1}{x} - \frac{1}{y} = \frac{y-x}{xy}\)) and the denominator (\(\frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy}\)).
- Substitute these back into the original fraction to get \(\frac{y-x}{xy} \div \frac{y+x}{xy}\).
- Simplify the division by multiplying by the reciprocal: \(\frac{y-x}{xy} \times \frac{xy}{y+x} = \frac{y-x}{y+x}\).
- The final equivalent expression is \(\boxed{\frac{y-x}{y+x}}\).
Explanation
- Problem Analysis
We are tasked with simplifying the complex rational expression:
\[ \frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}} \]
For this expression to be defined, the denominators \(x\) and \(y\) must be non-zero, and the denominator of the main fraction \(\frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy}\) must not be zero, meaning \(y + x
eq 0\). Our goal is to manipulate this complex fraction into a simpler quotient of polynomials.
- Simplifying the Numerator
First, let's simplify the numerator of the expression:
\[ \frac{1}{x} - \frac{1}{y} \]
To combine these fractions, we find a common denominator, which is \(xy\). Multiplying the first term by \(\frac{y}{y}\) and the second term by \(\frac{x}{x}\), we get:
\[ \frac{1 \cdot y}{x \cdot y} - \frac{1 \cdot x}{y \cdot x} = \frac{y}{xy} - \frac{x}{xy} = \frac{y - x}{xy} \]
- Simplifying the Denominator
Next, we simplify the denominator of the original expression in the same way:
\[ \frac{1}{x} + \frac{1}{y} \]
Using the same common denominator \(xy\), we have:
\[ \frac{1 \cdot y}{x \cdot y} + \frac{1 \cdot x}{y \cdot x} = \frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy} \]
- Final Simplification
Now, we substitute these simplified expressions back into the original main fraction:
\[ \frac{\frac{y - x}{xy}}{\frac{y + x}{xy}} \]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[ \frac{y - x}{xy} \cdot \frac{xy}{y + x} \]
Notice that the factor \(xy\) appears in both the numerator and the denominator, so they cancel out:
\[ \frac{y - x}{\cancel{xy}} \cdot \frac{\cancel{xy}}{y + x} = \frac{y - x}{y + x} \]
Thus, the simplified form of the given expression is \(\frac{y - x}{y + x}\).
Examples
In fields like electrical engineering, complex circuits with resistors in parallel often require calculations involving fractions of resistances. Suppose you have two resistors \(R_1\) and \(R_2\) connected in a way that the total resistance is determined by a formula involving \(\frac{1}{R_1}\) and \(\frac{1}{R_2}\). By simplifying these complex expressions, engineers can quickly find effective resistance values, which helps in optimizing power distribution and ensuring that circuits operate safely within their design parameters.