Solving for 1/(x*y) Given x3y335 and xy5

In this article, we will walk through the process of solving a challenging algebraic problem. Given the equations x3y335 and xy5, we will find the value of 1/x1/y.

Step-by-Step Solution

First, let us start with the initial equation:

xy335

Since xy5, we can substitute x3y3(xy)xy55.

Re-writing, we get:

x3y335

Renaming xy35xy:

xy3xy#x03A7;xy

Or,

x3y335

Thus, we get:

3555?15xy

Dividing both sides by 5, we obtain:

725?3xy

Therefore, we have:

3xy18

Thus, solving for xy gives:

xy6

Finally, we need to solve for 1/x1/y. Using the value of xy obtained, we can write:

1/x1/yxyxy5/6

Hence, the final answer is:

5/6

Conclusion

Through careful algebraic manipulation, we are able to solve for an indeterminate form involving the reciprocal of the product of two variables. By leveraging known relationships and simplifying the equation, we have successfully found the solution to the given problem.

Key Takeaways

Algebraic simplification is crucial in solving complex equations. Using substitution and known values can greatly simplify the process. The application of basic arithmetic operations can lead to the solution of complex problems.

Understanding and applying these algebraic techniques are fundamental in advancing your mathematical problem-solving skills.