The typical algebraic object the student runs into on the street is the equation. The following Properties of Equations provide us with the definition of what equations are in the context of algebra and CRAM Math.

There are other algebraic objects, such as inequalities, systems of equations and systems of inequalities. Their properties are also important for a complete understanding of algebra, but for now, we will concentrate on understanding equations first.

Reflexive Property of Equations

Any quantity a is equal to itself.

a = a

Symmetric Property of Equations

If the left hand expression a of an equation is equal to the right hand expression b, then b is equal to a.

(a = b) \to (b = a)

Transitive Property of Equations

If a quantity a is equal to another quantity b, and the second quantity b is equal to a third quantity c, then the first quantity a is equal to the third quantity c.

(a = b) \land (b = c) \to (a = c)

Substitution Property of Equations

If a quantity a is equal to b, then b replaces every instance of a written in an expression E.

(a = b) \to (E(a) \to E(b))

Additive Property of Equations

If a is equal to b, then the sum of a and a third quantity c is equal to the sum of b and c.

(a = b) \to (a + c = b + c)

If a is equal to b, then the product of a and a third quantity c is equal to the product of b and c

(a = b) \to (a \times c = b \times c)
(a = b) \to (ac = bc)

How to use

Remembering our rule of inference modus ponens, we may use a logical deduction to act as a re-write rule, combining known facts (complex equation) into some new fact whenever we have an algebraic argument of the form A \to B; A; \therefore B.


  1. y = mx + b (given)
  2. (y = mx + b) \to (mx + b = y) (instance of Symmetric Property)
  3. \therefore mx + b = y (MP, 1, 2)

This is a trivial example, but it illustrates the concept of using the properties that are in the form A \to B as re-written lines of an assigned problem, leading to the conclusion B according to the particular property you happen to choose to use.

Notice how the limited number of properties also limits the number of re-writing possibilities you have, guiding you from one step to another in making progress to a solution.

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