evaluating by substitution

An algebraic expression is like a recipe with blanks: it tells you what to do, but not with which numbers. Evaluating by substitution means filling in the blanks — you replace each variable with a given number, then carry out the arithmetic to get a single value.

The method has two steps. First, substitute: put the given value in place of every occurrence of the variable, usually in parentheses to protect signs. Then evaluate: simplify using the order of operations. For 3x - 2 at x = 4, write 3(4) - 2 = 12 - 2 = 10. The same expression yields different values for different inputs.

Parentheses around the substituted value are not optional decoration — they prevent sign errors. To evaluate -x^2 at x = -3, write -(-3)^2 = -(9) = -9, not 9: the exponent applies to -3 first, and the leading minus stays outside. Skipping the parentheses is the classic way to get the wrong sign.