Variables and Expressions

Objective: Distinguish unknown or varying quantities from fixed numbers.

A variable is a symbol standing for a number in a specified domain. In 3x+5, x is a variable, 3 is its coefficient, and 5 is a constant term. A variable may represent an unknown to solve for or an input allowed to change. A term is a number, variable, or product of factors separated by addition or subtraction. The expression 3x+5 has two terms, not an equals sign; without a given value for x it does not have one numerical value.

Objective: Translate quantities with order and grouping intact.

Words can specify arithmetic operations and grouping. “Five less than twice n” is 2n−5, while “twice the difference of n and five” is 2(n−5); they are unequal in general. A quotient requires care: “x divided by 4” is x/4 and “4 divided by x” is 4/x, with x≠0 for the latter. Translating context also requires units: if p is a price in dollars, 3p is the cost of three equal-price items in dollars. Define the variable before building a model.

Objective: Substitute and calculate using the given domain.

To evaluate an expression at a specified variable value, replace every occurrence of the variable, use parentheses for negative substitutions, and apply the order of operations. For 2x²−3 at x=−2, compute 2(−2)²−3=2·4−3=5. The square applies to the substituted negative value; −2² without parentheses means −(2²)=−4 by standard precedence. If an expression has a denominator, the substituted value cannot make it zero. In applications, interpret the computed value with the correct unit and a sensible domain.