Evaluate variable expressions involving rational numbers

Learning Objectives Substitute positive and negative rational numbers (fractions, decimals) into multi-variable expressions. Apply the correct order of operations (PEMDAS) to expressions involving rational numbers, exponents, and grouping symbols. Simplify complex fractions that result from substituting rational values into expressions. Evaluate expressions involving absolute values with rational number inputs. Accurately compute expressions with mixed operations (addition, subtraction, multiplication, division) on rational numbers. Interpret the result of an evaluated expression in the context of a given formula from science or finance. Ever used a physics formula like `d = v_0 t + (1/2)at^2` and had to plug in fractional time or acceleration? ⚛️ That's exactly what we...

TermDefinitionExample Variable ExpressionA mathematical phrase that contains variables, constants, and operation symbols. Its value changes depending on the values assigned to the variables.The expression `(3/4)x^2 - 2y` contains variables `x` and `y`, constants `3/4` and `2`, and operations of exponentiation, multiplication, and subtraction. Rational NumberAny number that can be written as a fraction `p/q`, where `p` and `q` are integers and `q ≠ 0`. This includes integers, terminating decimals, and repeating decimals.-5, 7/3, 0.6 (which is 6/10), -4.25 (which is -17/4). SubstitutionThe process of replacing variables in an expression with their specified numerical values.In the expression `5a - b`, substituting `a = 1/2` and `b = -3/4` results in `5(1/2) - (-3/4)`. Order of Operations (P...

Substitution Principle If `a = b`, then `a` can replace `b` in any expression. This is the fundamental principle that allows us to evaluate expressions. Always use parentheses when substituting values, especially if they are negative or fractions, to preserve the order of operations. For example, substitute `x = -1/2` into `3x^2` as `3(-1/2)^2`. Operations with Rational Numbers For fractions `a/b` and `c/d`: Addition/Subtraction: `a/b ± c/d = (ad ± bc) / bd`. Multiplication: `(a/b) * (c/d) = ac / bd`. Division: `(a/b) / (c/d) = (a/b) * (d/c) = ad / bc`. These are the core rules for arithmetic with fractions. For addition and subtraction, you must find a common denominator. For division, you multiply by the reciprocal of the divisor.