Order of Operations and Simplifying Algebraic Expressions

Order of Operations and Simplifying Expressions

Objectives

This section introduces foundational concepts in algebra, focusing on evaluating algebraic expressions and using mathematical models. Mastery of these skills is essential for success in College Algebra.

• Evaluate algebraic expressions

• Use mathematical models

Algebraic Expressions

Definition and Components

An algebraic expression is a combination of variables and numbers connected by operations such as addition, subtraction, multiplication, division, powers, or roots. A variable is a letter used to represent various numbers. An exponential expression is an expression of the form $b^n$.

• Variable: A symbol (often a letter) representing a number whose value can change.

• Algebraic Expression: A mathematical phrase involving variables, numbers, and operations.

• Exponential Expression: An expression where a variable or number is raised to a power, e.g., $b^n$.

• Evaluating an Algebraic Expression: Finding the value of the expression for a given value of the variable.

Example: Evaluating an Algebraic Expression

To evaluate $8 + 6(x - 3)^2$ for $x = 13$:

• Substitute $x = 13$ into the expression:

• $8 + 6(13 - 3)^2$

• $8 + 6(10)^2$

• $8 + 6 imes 100$

• $8 + 600 = 608$

Checkpoint Example

Evaluate $2 + 3(x - 1)^2 - 4$ for $x = 3$:

• Substitute $x = 3$:

• $2 + 3(3 - 1)^2 - 4$

• $2 + 3(2)^2 - 4$

• $2 + 3 imes 4 - 4$

• $2 + 12 - 4 = 10$

Formulas and Mathematical Models

Equations, Formulas, and Modeling

An equation is formed when an equal sign is placed between two algebraic expressions. A formula is an equation that uses variables to express a relationship between two or more quantities. Mathematical modeling is the process of finding formulas to describe real-world phenomena. These formulas, along with the meaning assigned to the variables, are called mathematical models.

• Equation: $a = b$

• Formula: $A = rac{1}{2}bh$ (area of a triangle)

• Mathematical Model: A formula representing a real-world relationship

Example: Using a Mathematical Model

The formula $T = 4x^2 + 341x + 3194$ models the average cost of tuition and fees, $T$, for public U.S. colleges for the school year ending $x$ years after 2000.

• To project the average cost for the year ending in 2015, substitute $x = 15$:

• $T = 4(15)^2 + 341(15) + 3194$

• $T = 4 imes 225 + 5115 + 3194$

• $T = 900 + 5115 + 3194 = 11209$

Additional info: The formula is a quadratic model, commonly used in economics and social sciences to represent growth over time.

Checkpoint Example

Find the average cost for the year ending in 2010 ($x = 10$):

• $T = 4(10)^2 + 341(10) + 3194$

• $T = 4 imes 100 + 3410 + 3194$

• $T = 400 + 3410 + 3194 = 7004$

Simplifying Algebraic Expressions

Combining Like Terms

To simplify an algebraic expression, combine like terms. Like terms have exactly the same variable factors. An expression is simplified when parentheses are removed and like terms are combined.

• Like Terms: Terms with the same variable(s) raised to the same power(s).

• Simplified Expression: An expression with all like terms combined and no parentheses.

Example: Simplifying an Algebraic Expression

Simplify $7(4x^2 + 3x) + 2(5x^2 + x)$:

• Distribute:

• $= 28x^2 + 21x + 10x^2 + 2x$

• Combine like terms:

• $= (28x^2 + 10x^2) + (21x + 2x)$

• $= 38x^2 + 23x$

Checkpoint Example

Simplify $6(2x^2 + 4x) + 10(4x^2 + 3x)$:

• Distribute:

• $= 12x^2 + 24x + 40x^2 + 30x$

• Combine like terms:

• $= (12x^2 + 40x^2) + (24x + 30x)$

• $= 52x^2 + 54x$

Checkpoint Answers

Checkpoint	Answer
CP 1	10
CP 2	$T = 7004$
CP 7	$52x^2 + 54x$