Algebraic Expressions and Linear Equations

The Building Blocks of Algebra: Terms, Expressions, and Order of Operations

Algebra uses symbols to describe mathematical relationships and solve problems. A variable is a letter or symbol that represents a number that may be unknown or may change. In contrast, a constant is a fixed numerical value that does not change. When numbers and variables combine through multiplication, they form terms. A term is a number, a variable, or a product of numbers and variables, which may have exponents. A coefficient is a numerical factor multiplied by a variable. For example, in the algebraic term 7x, 7 is the coefficient and x is the variable. A variable written as x has coefficient 1, while -x has coefficient -1.

It is essential to recognize the difference between an expression and an equation. An algebraic expression is a combination of variables, numbers, and operations that does not contain an equals sign. An equation, however, is a mathematical statement showing that two expressions are equal using an equals sign. An expression represents a quantity, whereas an equation states that two quantities have the exact same value.

Evaluating an algebraic expression means substituting specific numbers for variables and calculating the resulting value. Whenever you evaluate an expression, you must follow the order of operations. The order of operations requires performing operations inside parentheses first, followed by exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Careful substitution is critical when working with negative quantities. When substituting a negative number into an expression, put it in parentheses to help avoid sign errors. Exponents require extra attention: for any real number x, (-x)^2 equals x^2 and is nonnegative, while -x^2 means -(x^2) and is nonpositive. Keeping parentheses around substituted negative values prevents confusion between these two distinct structures.