College Algebra: Algebraic Expressions and Linear Equations Study Guide

Algebraic Expressions and Linear Equations

Evaluating Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. Evaluating an expression means substituting a given value for the variable(s) and performing the indicated operations.

• Definition: An algebraic expression is a combination of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division).

• Key Steps:

  1. Substitute the given value(s) for the variable(s).

  2. Follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example: Evaluate $x^2 - 6x + 5$ for $x = 7$.

  • Substitute: $7^2 - 6(7) + 5$

  • Calculate: $49 - 42 + 5 = 12$

Removing Parentheses in Algebraic Expressions

Removing parentheses often involves distributing a negative sign or a coefficient across the terms inside the parentheses.

• Key Point: When a negative sign precedes parentheses, change the sign of each term inside.

• Example: Write $-(6x - 9y - 6)$ without parentheses.

  • Distribute the negative: $-6x + 9y + 6$

Simplifying Algebraic Expressions

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

• Key Steps:

  1. Distribute any coefficients across parentheses.

  2. Combine like terms (terms with the same variable and exponent).

• Example: Simplify $8(6y - 4) - (5y + 7)$.

  • Distribute: $8 imes 6y = 48y$, $8 imes -4 = -32$

  • Expression becomes: $48y - 32 - 5y - 7$

  • Combine like terms: $48y - 5y = 43y$, $-32 - 7 = -39$

  • Final answer: $43y - 39$

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving a linear equation involves finding the value of the variable that makes the equation true.

• Definition: A linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Key Steps:

  1. Expand and simplify both sides of the equation.

  2. Isolate the variable on one side.

  3. Solve for the variable.

• Example: Solve $9x-(5x-6)=26$.

  • Expand: $9x - 5x + 6 = 26$

  • Combine like terms: $4x + 6 = 26$

  • Subtract 6 from both sides: $4x = 20$

  • Divide by 4: $x = 5$

Summary Table: Key Operations with Algebraic Expressions

Operation	Example	Result
Evaluate	$x^2 - 6x + 5$ for $x=7$	$12$
Remove Parentheses	$-(6x - 9y - 6)$	$-6x + 9y + 6$
Simplify	$8(6y - 4) - (5y + 7)$	$43y - 39$
Solve Linear Equation	$9x - (5x - 6) = 26$	$x = 5$