Exploring Factorization Techniques in Solving Systems of Equations

System of equations can present a challenge, but using factorization techniques can simplify the process. This article demonstrates how to solve a given system of equations by utilizing algebraic manipulation and factorization. We will explore different methods, from basic factorization to more complex algebraic transformations, and understand the underlying principles.

Introduction to the Problem

We start with the following system of equations:

[ x^3 x^2 x^1 y^3 1 quad (1) ]

[ y^3 y^2 y^1 x^3 1 quad (2) ]

Let's examine these equations and find all possible pairs (x, y) that satisfy both equations.

Factorization and Simplification

First, we rewrite the equations in a more appealing form and then look for common factors.

[ x^3 x^2 x^1 y^3 1 quad text{and} quad y^3 y^2 y^1 x^3 1 ]

By multiplying and collecting like terms in both equations, we get:

[ x 1 y 1 x - y 1 - x y 0 ]

This equation simplifies to several possible relationships between x and y: x -1 y -1 x y xy 1

Case Analysis

We will now analyze these cases one by one.

Case 1: x y

If x y, we can substitute y in the first equation:

[ x^3 x^2 x^1 x^3 1 ]

This reduces to:

[ x^6 x^5 x^4 1 0 ]

The above equation can be factored and solved to give:

[ (x - 1)(x 1)^2 0 ]

Hence, x -1 or x 1. For x y -1 and x y 0, we get the pairs:

[ 0,0 quad text{and} quad -1, -1 ]

Case 2: xy 1

If xy 1, we substitute y 1/x in the first equation:

[ x^3 x^2 x^1 (1/x)^3 1 ]

This simplifies to:

[ x^6 x^5 x^4 1 0 ]

The above equation implies x -1, and thus y -1. This solution is consistent with the first case, giving us the pair:

[ -1, -1 ]

Conclusion

After analyzing all possible cases, we find that the solution pairs are:

[ (0,0), (-1,-1) ]

These are the only real solutions to the given system of equations.

Graphical Interpretation

The graphs of the two equations above can be seen as:

These graphs intersect at the trivial solutions (0,0) and (-1,-1). The symbolic manipulation of the equations and their graphical representation provide a richer understanding of the problem.