Solving Simultaneous Equations: A Step-by-Step Guide

Simultaneous equations can present a fascinating challenge for mathematicians and students alike. This article will guide you through the process of solving a specific set of simultaneous equations:

The Equations

We will solve the following set of equations:

[frac{x}{y} frac{y}{x} frac{26}{5}]
[x^2 - y^2 24]

Method 1: Trigonometric Substitution

To make the problem more manageable, we can use trigonometric substitution. Let:

[x sqrt{24} sec t]
[y sqrt{24} tan t]

Substituting these into the first equation and simplifying:

[frac{sec t}{tan t} frac{tan t}{sec t} frac{26}{5}]

This simplifies to:

[sec t tan t frac{26}{5}]

Welcome a step by step solution:

[sin^2 t frac{1}{5}]
[cos^2 t 1 - frac{1}{25} frac{24}{25}]

Solving for (x) and (y), we get:

[x frac{sqrt{24}}{cos t} pm 5]
[y pm 1]

The solutions are (x 5, y 1) and (x -5, y -1).

Method 2: Direct Substitution and Simplification

We can solve the equations directly by substituting and simplifying step by step:

[frac{x}{y} frac{y}{x} frac{26}{5}]

This can be rewritten as:

[frac{x^2 y^2}{xy} frac{26}{5}]

Multiplying both sides by (xy):

[x^2 y^2 frac{26}{5}xy]

And we know:

[x^2 - y^2 24]

Adding the equations:

[2x^2 frac{26}{5} xy 24]

Solving for (y):

[y frac{2x^2 - 24}{frac{26}{5}x}]

Substituting (y frac{2x^2 - 24}{frac{26}{5}x}) into (x^2 - y^2 24):

[x^2 - left(frac{2x^2 - 24}{frac{26}{5}x}right)^2 24]

Simplifying further, we get a quadratic in (x^2):

[x^4 - 24x^2 - 25 0]

Solving this quadratic equation:

[x^2 25 pm i^2]

The real solutions are (x pm 5), and substituting back we get (y pm 1).

Method 3: Direct Squaring and Simplification

Another approach is to square the first equation:

[x^2 y^2 2xy left(frac{26}{5}right)^2]

And knowing:

[x^2 - y^2 24]

Adding the equations:

[2x^2 frac{26}{5}xy 24]

Substituting (x^2) in terms of (y^2):

[2x^2 frac{26}{5} sqrt{x^2 - y^2} 24]

Finally, solving for (x) and (y), we get the same solutions:

[x pm 5]
[y pm 1]

Conclusion

There are multiple ways to solve these simultaneous equations, each with its own steps and simplifications. The real solutions are:

[x 5, y 1]
[x -5, y -1]

These methods showcase the beauty and complexity of solving simultaneous equations.