Reducing Linear Combinations to Symmetric Forms: A Study in Linear Transformations of Bivariate Expressions
Dear Editors,
This paper is an original work developed independently as part of my undergraduate exploration in mathematics at Pachhunga University College. I aimed to study how certain linear expressions can be transformed into symmetric forms and formulated conjectures and a theorem supported by examples and proof.
Please note that this is my first formal submission of academic work. I welcome any suggestions or corrections that would help improve the paper further.
Thank you for your consideration.
Sincerely,
Vanlalpeka Sailo
Abstract
This paper investigates when linear combinations of two expressions in the form
E1 = a1x + b1y and E2 = a2x + b2y can be transformed into a symmetric expression of
the form c(x +y), where c is a constant. We present a general theorem that guarantees
the existence of a rational scalar m such that mE1 ±E2 = c(x +y) under the condition
that a1 ̸= b1. The paper includes several conjectures supported by examples, exploring
special cases such as when the coefficients are prime or have distinct common factors.
This study is motivated by a desire to understand how basic algebraic expressions can
be manipulated into symmetric forms and to contribute to undergraduate mathematical exploration