Graphs & the Rectangular Coordinate System

Rectangular (Cartesian) Plane

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). This system is fundamental for graphing equations and visualizing relationships between variables in algebra.

• Horizontal axis: x-axis

• Vertical axis: y-axis

• Origin: The point (0, 0) where the axes intersect

• Ordered pairs: (x, y) indicate position on the plane

• Quadrants: The axes divide the plane into four quadrants, numbered counterclockwise starting from the top right

Example: Plot the points A (4, 3), B (–3, 2), C (–2, –3), D (5, –4), E (0,0), F (0, –3) on the graph.

Example: Graph the points W (1, –2), X (5, 2), Y (–3, –2), Z (–4, 3) and identify their quadrants.

Linear Expressions and Linear Equations

Definitions and Properties

A linear expression is an algebraic expression of the form ax + b, where a and b are constants. A linear equation is a linear expression set equal to another value, typically written as ax + b = c.

• Linear Expression:

• Linear Equation:

• Solving for known x: Substitute and simplify

• Solving for unknown x: Find the value(s) of x that make the equation true

Example: Identify and perform the operation needed to isolate x in equations such as x + 2 = 0 and 3x = 12.

Steps for Solving Linear Equations

• Distribute constants

• Combine like terms

• Group terms with x and constants on opposite sides

• Isolate x (solve for x)

• Check solution by substituting x in the original equation

Example: Solve

Linear Equations with Fractions

Solving Linear Equations Involving Fractions

Linear equations may contain fractions. To solve these, it is often helpful to eliminate fractions by multiplying both sides by the Least Common Denominator (LCD).

• Multiply by LCD to eliminate fractions

• Distribute constants

• Combine like terms

• Group terms with x and constants on opposite sides

• Isolate x

• Check solution by substituting x in the original equation

Example: Solve

Practice: Solve

Solving Linear Equations: Categorizing Solutions

Types of Solutions

Linear equations can be categorized based on the number of solutions:

• Identity: True for all real numbers (infinite solutions)

• Conditional: True for specific value(s) of x (one solution)

• Inconsistent: No solution

Practice: Solve and categorize the solution.

Rational Equations

Definition and Solution Restrictions

A rational equation is an equation in which a variable appears in the denominator of a fraction. Solutions cannot be any value that makes a denominator zero; these are called restrictions.

• Set denominator equal to zero to find restrictions

• Multiply both sides by LCD to eliminate fractions

• Solve the resulting linear equation

• Check solution against restrictions

Example: Solve ,

Practice: Solve

Solution Equal to Restriction

If the solution to a rational equation is equal to a restriction, then there is no solution.

Example: Solve ,

Square Roots of Negative Numbers & Imaginary Numbers

Imaginary Unit

Square roots of positive numbers are real, but square roots of negative numbers are not real. The imaginary unit i is defined as .

• To simplify , write as where b is positive

Example:

Example:

Note: All such solutions are called imaginary numbers.

Powers of i

Evaluating Powers of the Imaginary Unit

All properties of exponents apply to powers of i. The powers of i repeat in a cycle of four:

Any power of i can be simplified to one of these four values.

• To evaluate , divide n by 4 and use the remainder to determine the value.

Example: , ,

Introduction to Complex Numbers

Standard Form and Components

Complex numbers combine real and imaginary parts and have the standard form , where a is the real part and b is the imaginary part.

• Real part: a

• Imaginary part: b

Example: For , a = 4, b = -3

Practice: Write in standard form.

Adding & Subtracting Complex Numbers

Combining Like Terms

To add or subtract complex numbers, combine the real parts and the imaginary parts separately. Always express the answer in standard form.

Example:

Practice:

Multiplying Complex Numbers

Distributive Property and FOIL

Complex numbers are multiplied using the distributive property or FOIL method. Remember that .

• Distribute or FOIL

• Apply

• Combine like terms

Example:

Example:

Practice:

Practice:

Additional info: The study guide covers foundational topics in College Algebra, including graphing, solving linear and rational equations, and operations with complex numbers. It provides definitions, examples, and step-by-step procedures suitable for exam preparation.