Ch. 6: Equations and Inequalities 6.1 Algebraic Expressions and Formulas

Objectives Evaluate algebraic expressions. Use mathematical models. Understand the vocabulary of algebraic expressions. Simplify algebraic expressions.

Algebraic Expressions An algebraic expression: a combination of variables and numbers with operators of addition, subtraction, multiplication, or division as well as powers or roots. Example:

Order of Operations Given an expression involving many terms and operators: Evaluate parentheses from inside out. Evaluate exponential expressions. Perform multiplications and divisions, from left to right. Perform additions and subtractions, from left to right. Evaluate: 7 + 5 (x – 4)3 for x = 6

Solution Let x = 6. 7 + 5 (x – 4)3 = 7 + 5(6 – 4)3 = 7 + 5(2)3 = 7 + 5(8) = 7 + 40 = 47

Terminology An equation: Statement of equality between two algebraic expressions E.g., 2x + 3y = 4 A formula: Equation that uses letters to express a relationship between two or more variables. E.g., C = (5/9)(F – 32) Mathematical modeling: Process of finding formulas to describe real-world phenomena. E.g., in finance: M = P( 1 + i )n M is the final amount including the principal. P is the principal amount. i is the rate of interest per year. n is the number of years invested.

Example: Modeling Caloric Needs The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. The mathematical model W = 66x2 + 526x + 1030 describes the number of calories needed per day by women in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by women between the ages of 19 and 30, inclusive, with this lifestyle?

Solution Solution: For ages 19-30 (group4), x = 4 The formula indicates that 2078 calories are needed per day by women in the 19-30 age range with moderately active lifestyle.

Evaluating a Formula Initial height: 4 ft Initial speed: 60 ft/sec Formula: h = 4 + 60t – 16t2 What is the ball’s height 2 sec after it was kicked? h = 4 + 60(2) – 16(22) = 4 + 120 – 64 = 60 What is the ball’s height 3 sec after it was kicked? h = 4 + 60(3) – 16(32) = 4 + 180 – 144 = 40

Example: Body Mass Index mass(lb) Formula: BMI = ------------------ x 703 (height (in))2 BMI Calculator BMI Table

Terminology Term: Those parts of an algebraic expression separated by addition/subtraction operators. E.g.: in the expression 7x – 9y – 3 Coefficient: The numerical part of a term. 7, –9, –3 Constant: A term that consists of just a number, also called a constant term. –3 Like terms: Terms that have the exact same variable factors. 7x and 3x Factors: Parts of each term that are multiplied. E.g.: In 5x(y – z), 5, x and (y – z) are factors.

Simplifying Algebraic Expressions Simplify: 5(3x – 7) – 6x Solution: 5(3x – 7) – 6x = 5∙3x – 5∙7 – 6x = 15x – 35 – 6x = (15x – 6x) – 35 = 9x – 35

Your Turn 5x(x – 3) – 2(x + 4) 3xy(z + 3) – (xyz + xy) Simplify: 5x(x – 3) – 2(x + 4) = 5x2 – 15x – 2x – 8 = 5x2 – 17x – 8 3xy(z + 3) – (xyz + xy) = 3xyz + 9xy – xyz – xy = 2xyz + 8xy