Final answer:
To simplify the complex fraction ((1/x) + (x/y)) / ((1/y) + 1), one must find a common denominator, multiply both the numerator and the denominator by this, and then simplify. The result contains terms that may allow for further simplification or provide a means to solve for a variable if needed.
Step-by-step explanation:
Simplifying the complex fraction ((1)/(x) + (x)/(y)) / ((1)/(y) + 1) involves a few steps. First, to eliminate the complexity, you can multiply both the numerator and the denominator by a common denominator that would eliminate the individual fractions within the complex fraction. In mathematics, when we multiply the numerators and the denominators by the same thing, this is like multiplying by 1, which means the expression is still an equality. In this case, the common denominator would be xy. Multiplying the numerator by xy, you get xy * (1/x + x/y) = y + x^2 . Multiplying the denominator by xy, you end up with xy * (1/y + 1) = x + xy. This gives you the simplified expression (y + x^2) / (x + xy).
Further simplification can be done by factoring out common terms. If you wanted to solve for y, you would need to manipulate the equation further. Any time the same quantity appears both in a numerator and a denominator, they cancel out, simplifying the fraction.
Remember, when simplifying and performing operations on both sides of the equals sign, the expression remains an equality. A good example to grasp this concept is to note that a fraction with the same number in both the numerator and denominator is equal to 1. In your final answer, no such simplification may occur, but this rule is important when simplifying individual fractions before finding a common denominator.