GMAT Exam: A Simple Guide To Algebra - Algebraic Expressions

Basic Concepts
A variable holds a place open for a number (or numbers, in some cases) whose value may vary. Upper or lowercase letters (e.g., x, y, z, A, B, or C) represent variables.

For simplicity, the letter is the “name” of the variable.

In problem solving, you use variables to represent unknown quantities. Although a variable may represent any number, in many problems, the variables represent specific numbers, but the values are unknown.

A constant is a quantity that has a fixed, definite value that does not change in a problem situation.

For example, all the real numbers are constants, including real numbers whose units are units of measure such as 5 feet, 60 degrees, 100 pounds, and so forth.

The result of multiplying 5 and x is 5x. Thus, 5 · x = x · 5 = (5)(x) = (5)x = x(5) = 5x.

A constant factor times a variable is the variable’s numerical coefficient.

For example, 5 is the numerical coefficient of x in the expression 5x + 3.

A variable for which a numerical factor is not shown has the numerical coefficient of 1.

For example, the numerical coefficient of x in the expression x + 3 is 1.

The numerical coefficient of a variable that is multiplied by several factors is their product.

For example, the numerical coefficient of x in the expression (5)(4)x + 10 is (5)(4) = 20.

An algebraic expression is a symbolic representation of a number. It can contain constants, variables, and computation symbols.

Multiplication is shown by writing variables and coefficients or two or more variables (with or without constants) side by side with no multiplication symbol.

Thus, –5x means –5 times x, and 2xyz means 2 times x times y times z.

Also, a number or variable written immediately next to a grouping symbol indicates multiplication.

For instance, 6(x + 1) means 6 times the quantity (x + 1), means 7 times , and –1|–8| means –1 times |–8|.

Evaluating Algebraic Expressions
If you are given numerical values for the variables in an algebraic expression, you can evaluate the expression by substituting the given numerical value for each variable and then simplifying by performing the indicated operations, being sure to follow the order of operations as you proceed.

For instance, when x = 4, y = –8, and z = –5:


When you substitute negative values into an algebraic expression, enclose them in parentheses to avoid careless errors.