Linear Equations and Their Forms

Standard Form of a Linear Equation

The standard form of a linear equation in two variables is given by:

• Ax + By = C, where A and B are real numbers and not both zero.

• This form is useful for quickly finding the x-intercept and y-intercept of the line.

• The values of A, B, and C can also be used to determine the slope of the line.

Intercepts of a Line

The intercepts are the points where the line crosses the axes:

• x-intercept: Set y = 0 and solve for x.

• y-intercept: Set x = 0 and solve for y.

Forms of Linear Equations

There are three main forms of linear equations:

• Standard Form:

• Slope-Intercept Form:

• Point-Slope Form:

Classification of Lines by Slope

Lines can be classified based on their slope:

• Slanted Line: (A, B ≠ 0) - can have positive or negative slope.

• Horizontal Line: (k is a constant) - slope is zero.

• Vertical Line: (k is a constant) - slope is undefined.

Graphing Linear Equations

Graphing from Standard Form

To graph a linear equation in standard form:

1. Find the x-intercept by setting y = 0.

2. Find the y-intercept by setting x = 0.

3. Plot both intercepts and draw a straight line through them.

Examples

Slope of a Line

Definition and Formula

The slope of a line measures its steepness and is calculated using two points and :

• Slope formula:

• Also described as "rise over run" or the change in y divided by the change in x.

• If , the slope is undefined (vertical line).

Examples

• Find the slope between and :

• Find the slope between and :

• For , the slope is 0 (horizontal line).

• For , the slope is undefined (vertical line).

Point-Slope Form

The point-slope form is useful for writing the equation of a line given a point and the slope:

Slope-Intercept Form

Definition

The slope-intercept form of a line is:

• Where m is the slope and b is the y-intercept.

Example

• Write the equation of a line passing through with slope 5: (find b using the point).

Average Rate of Change

Definition and Formula

The average rate of change of a function on the interval is the slope of the secant line connecting and :

• This measures how the function changes on average between two points.

Examples

• From to

• From to

Solving Linear Equations

Example Equation

• Solve

• Steps:

  1. Move all terms involving x to one side:

  2. Simplify:

  3. Divide:

Summary Table: Linear Equations and Slope Types

The following table summarizes the types of linear equations and their slopes:

Key Points to Remember

• Two points determine a unique line.

• The slope is undefined for vertical lines.

• The slope is zero for horizontal lines.

• Linear equations can be written in standard, slope-intercept, or point-slope form.

• The average rate of change is a fundamental concept for understanding how functions behave between two points.