Algebraic Expressions and Basic Operations: College Algebra Study Notes

Algebraic Expressions and Basic Operations

Introduction to Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. Understanding how to manipulate and evaluate these expressions is foundational in College Algebra.

• Expression: A combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and division).

• Variable: A symbol, usually a letter, that represents an unknown value.

• Constant: A fixed value.

Subtraction and Addition of Integers

Adding and subtracting integers is a critical skill required for manipulating algebraic expressions.

• Addition: To add two numbers, find the sum.

• Subtraction: To subtract, find the difference between them.

• Example: $7 - 3 = 4$

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given value for the variable and perform the indicated operations.

• Substitute values: Replace each variable with its given value.

• Order of Operations: Follow Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

• Example: If $x = 2$, evaluate $2x + 3$:

$2x + 3 = 2(2) + 3 = 4 + 3 = 7$

Simplifying Expressions

Simplifying expressions involves combining like terms and performing arithmetic operations to write the expression in its simplest form.

• Like terms: Terms that have the same variable raised to the same power.

• Example: $2x + 3x = (2 + 3)x = 5x$

Key Properties of Operations

• Commutative Property: $a + b = b + a$ and $ab = ba$

• Associative Property: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$

• Distributive Property: $a(b + c) = ab + ac$

Sample Table: Order of Operations

Step	Operation	Example
1	Parentheses	$2(p + 3)$
2	Exponents	$p^2$
3	Multiplication/Division	$2x$
4	Addition/Subtraction	$2 + x$

Practice Problems

• Subtract: $8 - 11$

• Simplify: $2x + 3x$

• Evaluate: $2x + 3$, if $x = 4$

• Simplify: $2a + 3b - 2a + 4b$

• Evaluate: $x^2 + y$, if $x = 3$, $y = 4$

• Subtract: $-11 - 7$

• Find the difference: $12 - 5$

Additional info: These problems and explanations cover foundational skills in algebraic manipulation, relevant to Chapter 2: Equations, Inequalities, and Problem Solving in College Algebra.