Solving Systems of Equations: Techniques and Solutions

In mathematics, solving systems of equations is a fundamental problem that is encountered in various fields, from physics to engineering. This article focuses on a specific system of equations and demonstrates different methods to find its solutions. By exploring both analytical and graphical approaches, we can gain a deeper understanding of the underlying mathematical concepts.

Introduction to the System of Equations

Consider the following system of equations:

sin(x)sin(y) sin(xy)

At first glance, this equation might seem complex and daunting. However, with the right approach, we can uncover its solutions step-by-step.

Graphical Approach

A common and intuitive method to solve such equations is by using graphing software. By plotting the functions on a coordinate plane, we can visually identify the points where the graphs intersect. These intersection points correspond to the solutions of the equations. In our example, the plot reveals six distinct solutions, which are:

(10, 01), (0-1, 1/2), (-1/2, 1/2), (-1/2, -1/2), (1/2, 1/2), (1/2, -1/2)

Analytical Approach

While the graphical method is useful for visualization, an analytical approach provides a deeper understanding of the underlying mathematical principles. Let's break down the equation step-by-step:

Step 1: Simplification of the Equation

We start with the given equation:

sin(x)sin(y) sin(xy)

By applying the trigonometric identity for the sine of a product, we can rewrite the equation as:

2sin(frac{xy}{2})cos(frac{x-y}{2}) 2sin(frac{xy}{2})cos(frac{xy}{2})

This can be simplified to:

sin(frac{xy}{2})[cos(frac{x-y}{2}) - cos(frac{xy}{2})] 0

Therefore, we have two cases:

sin(frac{xy}{2}) 0

OR

cos(frac{x-y}{2}) - cos(frac{xy}{2}) 0

Case 1: sin(frac{xy}{2}) 0

This implies:

frac{xy}{2} npi, n in mathbb{Z}

xy 2npi, n in mathbb{Z}

Thus, we have the set of solutions for this case:

(xy 2npi, n in mathbb{Z})

Case 2: cos(frac{x-y}{2}) - cos(frac{xy}{2}) 0

This simplifies to:

cos(frac{x-y}{2}) cos(frac{xy}{2})

frac{x-y}{2} pm frac{xy}{2} 2kpi, k in mathbb{Z}

Given that sin(xy) 1, we consider the specific case:

xy 2pi n, n in mathbb{Z}

This further implies:

2x 1, x frac{1}{2}

xy 0, x 0, y 1

xy 1, x leq 1, y leq 1, xy leq 1

Conclusion

By combining both graphical and analytical methods, we can solve the given system of equations effectively. The graphical approach provides a visual understanding, while the analytical method offers a deeper insight into the underlying mathematical principles. Understanding these techniques not only helps in solving specific equations but also enhances problem-solving skills in mathematics and related fields.