Linear Equations in Two Variables

Standard Form of a Linear Equation

A linear equation in two variables can be written in the form , where A, B, and C are real numbers and A and B are not both zero. This form is called standard form.

• Standard Form:

• Variables: x and y

• Coefficients: A, B, C (real numbers)

Intercepts

Two useful points for graphing are the x-intercept and y-intercept. The x-intercept is found by setting , and the y-intercept is found by setting .

• x-intercept: Let and solve for .

• y-intercept: Let and solve for .

Example: Finding Intercepts and Graphing

Find the x- and y-intercepts and graph the equation .

• x-intercept:

• y-intercept:

Special Cases: Equations with One Variable

Horizontal and Vertical Lines

If the equation to be graphed is "missing" a variable, it represents a special case:

• Horizontal line: (missing x)

• Vertical line: (missing y)

Graphing Linear Equations

Graphing by Table or Slope-Intercept Form

Linear equations can be graphed by making a t-table (table of values) or by rewriting the equation in slope-intercept form ().

• Slope-intercept form:

• Slope (m): The rate of change of y with respect to x

• y-intercept (b): The value of y when x = 0

Graphing Lines Through the Origin

If a linear equation in standard form has a C value of 0, then the line will go through the origin.

• Form: or

Midpoint Formula

Definition and Formula

The midpoint of a line segment is the point that is halfway between the endpoints. If the endpoints are and , the midpoint is:

• Midpoint Formula:

Example: Finding the Midpoint

Find the coordinates of the midpoint of the line segment with endpoints and .

• Step 1: Average the x-values:

• Step 2: Average the y-values:

• Midpoint:

Slope of a Line

Slope Formula

The slope of a line through two distinct points and is given by:

• Slope Formula:

• Interpretation: Slope is the "rise over run" or the change in y divided by the change in x.

Example: Calculating Slope

Find the slope of the line through the points and .

• Step 1:

• Interpretation: The slope is negative, indicating the line decreases as x increases.

Special Slope Cases

• Undefined slope: Vertical lines ()

• Zero slope: Horizontal lines ()

Summary Table: Types of Linear Equations

Additional info: The notes also include graphical representations and step-by-step calculations for intercepts, slope, and midpoint, reinforcing the algebraic concepts with visual aids.