The letters in algebraic equations are referred to as unknowns. Thus, x is the unknown in the equation 3x + 5 = 8. Algebraic equations can have any number of unknowns. The name unknown arises because letters are substituted for the numerical values that are not known in a problem.

The number of unknowns in a problem determines the number of equations needed to solve for the numerical values of the unknowns. Problems involving one unknown can be solved with one equation, problems involving two unknowns require two independent equations, and so on.

The degree of an equation depends on the power of the unknowns. The degree of an algebraic term is equivalent to the exponent of the unknown. Thus, the term 3x is a first degree term; 3x² is a second degree term, and 3x³ is a third degree term. The degree of an equation is the same as the highest degree term. Linear or first degree equations contain no terms higher than first degree. Thus, 2x + 3 = 9 is a linear equation. Quadratic or second degree equations contain up to second degree terms, but no higher. Thus, x² + 3x = 6, is a quadratic equation. Cubic or third degree equations contain up to third degree terms, but no higher. Thus, 4x³ + 3x = 12 is a cubic equation.

The degree of an equation determines the number of roots of the equation. Linear equations have one root, quadratic equations have two roots, and so on. In general, the number of roots of any equation is the same as the degree of the equation.

Exponential equations are those in which the unknown appears in the exponent. For example, e-z." = 290 is an exponential equation. Exponential equations can be of any degree.

The basic principle used in solving any algebraic equation is: any operation performed on one side of an equation must also be performed on the other side for the equation to remain true. This one principle is used to solve all types of equations.

There are four axioms used in solving equations:

Axiom 1. If the same quantity is added to both sides of an equation, the resulting equation is still true.

Axiom 2. If the same quantity is subtracted from both sides of an equation, the resulting equation is still true.

Axiom 3. If both sides of an equation are multiplied by the same quantity, the resulting equation is still true.

Axiom 4. If both sides of an equation are divided by the same quantity, except 0, the resulting equation is still true.

Axiom 1 is called the addition axiom; Axiom 2, the subtraction axiom; Axiom 3, the multiplication axiom; and Axiom 4, the division axiom. These four axioms can be visualized by the balancing of a scale. If the scale is initially balanced, it will remain balanced if the same weight is added to both sides, if the same weight is removed from both sides, if the weights on both sides are increased by the same factor, or if the weights on both sides are decreased by the same factor.