Graphing Linear Equations

Linear equations create straight lines when graphed on a coordinate plane. These equations demonstrate the relationship between x and y coordinates of all points on the line.

Definition: A linear equation is an equation whose graph forms a straight line, with all points on the line being solutions to the equation.

Example: For the equation y = x + 1:

• Point $-1,0$ is a solution
• Point (0,1) is a solution
• Point (2,3) is a solution

Highlight: Special cases of linear equations include:

• Vertical lines (x = constant)
• Horizontal lines (y = constant)

Example: The equation y = -2x + 1 can be graphed by plotting points:

• When x = -1, y = 3
• When x = 0, y = 1
• When x = 2, y = -3

Understanding Slope of a Line

The slope of a line measures its steepness and direction, providing crucial information about the relationship between variables in linear equations.

Definition: Slope $m$ is the ratio of the change in y (rise) to the change in x (run) between any two points on a line.

Vocabulary:

• Rise: The vertical change between two points
• Run: The horizontal change between two points

Highlight: Different types of slopes indicate different line behaviors:

• Positive slope: Line rises from left to right
• Negative slope: Line falls from left to right
• Zero slope: Horizontal line
• Undefined slope: Vertical line

Example: The slope formula is represented as: m = (change in y)/(change in x) = $y₂ - y₁$/$x₂ - x₁$

Understanding the Coordinate Grid

The coordinate grid serves as the fundamental framework for graphing linear equations. It consists of two perpendicular axes - the x-axis (horizontal) and y-axis (vertical) - that intersect at the origin point (0,0).

Definition: The coordinate grid is a two-dimensional plane formed by horizontal and vertical lines where points are located using ordered pairs (x,y).

Vocabulary: Origin - The point where the x and y axes intersect, represented as (0,0).

Example: The point (2,3) on the coordinate grid represents a location 2 units right of the origin and 3 units up.