Solving Simultaneous Equations with Reciprocals: A Detailed Guide

When dealing with simultaneous equations involving reciprocals, it's essential to understand the process and methods to transform and solve them systematically. This article explores the methods for solving the following set of reciprocal equations:

The Given Equations

The equations to be solved are:

1/x   1/y  51/y - 1/x  1

Step-by-Step Solution

Step 1: Introduce Substitutions

To simplify the equations, let's define: u 1/x, v 1/y Substituting these into the given equations, we get:

u v 5 v - u 1

Step 2: Solve for u and v

Now, we can solve these simpler equations step by step.

From the first equation:

v 5 - u

Substitute v into the second equation:

5 - u - u 1

Simplify:

5 - 2u 1

Solving for u:

-2u 1 - 5

-2u -4

u 2

Find v:

Using v 5 - u:

v 5 - 2 3

Step 3: Convert Back to x and y

Recall that:

u 1/x implies; x 1/u 1/2 v 1/y implies; y 1/v 1/3

The final solution is:

x 1/2, y 1/3

Conclusion

The values of x and y that satisfy the given equations are:

boxed{(1/2, 1/3)}

Additional Considerations

It is important to note that such simultaneous equations may have no valid solutions under certain conditions. For instance:

In the case of a more complex equation such as: (1/x)(1/y) 5 1/y - 1/x 10

Following similar steps, substituting and solving, you may encounter invalid situations where x or y cannot be determined, such as:

1/x 0 (which is impossible as no value of x can make 1/x equal to 0) 1/x -1/x (which simplifies to 2 0, an impossible statement)

These scenarios highlight the importance of carefully checking for valid solutions and understanding the limitations of algebraic manipulations.

Learning to identify and handle such cases is crucial for mastering the art of solving complex simultaneous equations.