The properties of equality (or axioms of equality) are used to solve equations. It is important to note that these properties are NOT for simplifying expressions; they describe various things one can do to an equation, while maintaining the validity of the equal sign throughout the entire process. There are four basic properties of equality: Substitution property, Reflexive property, Symmetric property and Transitive property. In addition to these four basic properties, the Addition property of equality and Multiplication property of equality are essential for solving equations. Research more about the properties of equality; introduce and explain each property textually and mathematically.

Your response should include the following:
A clear Introduction including the history of the basic properties of equality (Hint: Peano axioms).
A mathematical definition of each of the properties.
A discussion on which properties are mainly used for solving the equations and which are primarily used for various proofs.
In-text citations for any sources that you use to support evidence/examples as well as a complete and properly formatted citations in the Works Cited page at the end.