Graphing Linear Equations in Two Variables and the Rectangular Coordinate System

Equations and Inequalities in Two Variables

Introduction

This section introduces the foundational concepts of graphing linear equations in two variables, focusing on the rectangular coordinate system, ordered pairs, and methods for graphing lines. These topics are essential for understanding relationships between variables in College Algebra.

The Rectangular Coordinate System

Definition and Structure

• Rectangular coordinate system: Consists of a horizontal number line (x-axis) and a vertical number line (y-axis) that intersect at the origin (0, 0).

• Ordered pair (x, y): Represents a point in the plane, where x is the horizontal coordinate and y is the vertical coordinate.

• Example: The ordered pair (3, -4) means x = 3 and y = -4.

Linear Equations in Two Variables

Definition and Standard Form

• Linear equation in two variables: An equation that can be written in the form $Ax + By = C$, where A, B, and C are real numbers, and A and B are not both zero.

• Solution: An ordered pair (x, y) that makes the equation a true statement when substituted.

Example

• Is (2, 1) a solution to $2x + y = 5$?

• Substitute: $2(2) + 1 = 4 + 1 = 5$

• Since $5 = 5$, (2, 1) is a solution.

Graphing Linear Equations by Plotting Points

Method and Example

• Choose any value for x and solve for y to find ordered pairs.

• Plot at least three points to graph the line.

Example

• Equation: $y = -x + 3$

• Let $x = 1$: $y = -(1) + 3 = 2$

• Let $x = 2$: $y = -(2) + 3 = 1$

• Let $x = -1$: $y = -(-1) + 3 = 4$

x	y
1	2
2	1
-1	4

Plot these points and draw a straight line through them to represent the equation.

Graphing by Plotting Intercepts

Intercepts

• x-intercept: The point where the line crosses the x-axis; has the form (a, 0).

• y-intercept: The point where the line crosses the y-axis; has the form (0, b).

Example

• Equation: $-3y - 2x = -6$

• Find y-intercept: Set $x = 0$

• $-3y - 2(0) = -6 \Rightarrow -3y = -6 \Rightarrow y = 2$

• y-intercept: (0, 2)

• Find x-intercept: Set $y = 0$

• $-3(0) - 2x = -6 \Rightarrow -2x = -6 \Rightarrow x = 3$

• x-intercept: (3, 0)

Intercept	Value
y-intercept	(0, 2)
x-intercept	(3, 0)

Plot both intercepts and draw a line through them to graph the equation.

Special Cases: Vertical and Horizontal Lines

Definitions and Properties

• Vertical line: The graph of $x = a$ (where a is any real number) is a vertical line through the point (a, 0).

• Horizontal line: The graph of $y = b$ (where b is any real number) is a horizontal line through the point (0, b).

Examples

• $x = 2$ is a vertical line through (2, 0).

• $y = -3$ is a horizontal line through (0, -3).

Solving for a Variable

Isolating Variables in Linear Equations

• To solve for x in $2x + 1 = 11$:

• Subtract 1 from both sides: $2x = 10$

• Divide both sides by 2: $x = 5$

Summary Table: Key Concepts

Concept	Definition	Example
Ordered Pair	Represents a point (x, y) in the plane	(3, -4)
Linear Equation	$Ax + By = C$	$2x + y = 5$
x-intercept	Where line crosses x-axis	(a, 0)
y-intercept	Where line crosses y-axis	(0, b)
Vertical Line	$x = a$	$x = 2$
Horizontal Line	$y = b$	$y = -3$

Additional info: These notes expand on the provided slides and images to ensure completeness and clarity for College Algebra students. All equations are presented in standard LaTeX format for mathematical accuracy.