Let us treat this subject with the seriousness it deserves. This is a formal treatise on the principles of elementary algebra.

Postulate 1: We accept the field axioms of the real numbers as our foundation for all operations.

Postulate 2: The principle of equivalence, represented by the equals sign, is the fundamental relation we seek to preserve.

Theorem 1: The Additive Property of Equality. For any equation, adding the same value to both sides preserves the equivalence.

Theorem 2: The Multiplicative Property of Equality. Multiplying both sides of an equation by the same non-zero value preserves the equivalence.

From these simple, powerful ideas, the entire edifice of solving linear equations is rigorously constructed.

We then extend these principles to systems of multiple equations and variables, seeking a unique solution that satisfies all conditions.

The study then progresses to non-linear structures, namely polynomials, and the operations upon them.

The Fundamental Theorem of Algebra posits that a polynomial of degree 'n' will have 'n' complex roots.

Factoring and the Quadratic Formula are thus shown to be methods for discovering these fundamental roots.