Evaluating Algebraic Expressions

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First of all, an expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. Bound variables are assigned values within the expression (they are for internal use) while free variables can take on values from outside the expression.

The important thing is that expressions do not have equal signs in them. Expresions appear to be algebraic phrases, whereas equations are more like sentences. Does that make sense? As a general rule, expressions can be simplified but only equations can be solved.

For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the expression may be undefined. Thus an expression represents a function whose inputs are the values assigned the free variables and whose output is the resulting value of the expression.

For example, the expression

evaluated for x = 10, y = 5, will give the answer of 15 when the substitution property is applied correctly.

In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. One of many famous formulae is Albert Einstein's E = mc² (see special relativity).

How to Evaluate Expressions

There are a few things you need to remember in order to evaluate expressions. The most important thing to remember is that evaluating expressions involves the use of the Order of Operations. If you can't remember how to use the order of operations, then maybe we can go back and visit that section and refresh our memory. The second thing we need to remember is the substitution property. The substitution property was discussed in the last section, in Properties of Algebra. Lastly, we need to remember what it means to combine like terms. If we can remember and do these three things, then evaluating expressions should be a cinch!