Question - Solving a Fractional Equation with Variables
Solution:
The given expression seems to be a mathematical fraction which is missing the operator (like addition, subtraction, multiplication, or division) between the two terms in the numerator. If we assume that the desired operation is addition (which is common), then the expression reads:\[ \frac{1}{x} + \frac{1}{y} = 1 \]To solve this equation, you would normally need additional information about either x or y, since there are infinite solutions to this equation without additional constraints. However, we can manipulate the equation to express y in terms of x or vice versa. Let's solve for y in terms of x:$$1 = \frac{1}{x} + \frac{1}{y}$$Subtract $$ \frac{1}{x} $$ from both sides to isolate $$ \frac{1}{y} $$:$$1 - \frac{1}{x} = \frac{1}{y}$$Now take the reciprocal of both sides to solve for y:$$y = \frac{1}{1 - \frac{1}{x}}$$To simplify further, you'd find a common denominator in the numerator:$$y = \frac{1}{\frac{x - 1}{x}}$$Now you can see that y is the reciprocal of $$ \frac{x - 1}{x} $$, so we flip the fraction:$$y = \frac{x}{x - 1}$$This gives us y in terms of x. Again, without specific values or additional equations, we cannot find numerical solutions for x and y. But this manipulation gives us a relationship between the two variables.
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