Linear Functions

Equation of a Line

Linear functions describe straight lines in the coordinate plane and are fundamental in algebra. The equation of a line can be written in several forms, each useful for different purposes.

• Standard Equation:

• Slope-Intercept Equation:

Standard Equation of a Line

The standard form of a line is , where A, B, and C are constants. This form is useful for finding intercepts and for solving applications involving two quantities.

• x-intercept: The point where the line crosses the x-axis. Set and solve for :

• y-intercept: The point where the line crosses the y-axis. Set and solve for :

• Slope:

Example: For :

• x-intercept:

• y-intercept:

• Slope:

Example: For :

• x-intercept:

• y-intercept:

• Slope:

Slope-Intercept Equation of a Line

The slope-intercept form is , where m is the slope and b is the y-intercept. This form is especially useful for graphing and interpreting linear relationships.

• Slope (m): The rate of change of the line.

• y-intercept (b): The value of when .

Example: Slope and y-intercept :

• Equation:

Example:

• Slope:

• y-intercept: $4$

Example:

• Rewrite in slope-intercept form:

• Slope:

• y-intercept:

Finding the y-Intercept from a Point and Slope

Given a slope and a point , the y-intercept can be found using:

Example: Find the equation of the line with slope $3(2, 7)$:

• Equation:

Equation of a Line Given Two Points

Given two points and , the slope is:

Once the slope is found, use one point to solve for in .

Example: Points and :

• Slope:

• Using :

• Equation:

Parallel and Perpendicular Lines

Understanding the relationship between slopes helps classify lines as parallel or perpendicular.

• Parallel Lines: Have the same slope ().

• Perpendicular Lines: Slopes are negative reciprocals ().

Example: and are parallel (same slope).

Example: and are perpendicular (slopes are negative reciprocals).

Example: Are and parallel, perpendicular, or neither?

• First line: (slope )

• Second line: (slope )

• They are neither parallel nor perpendicular.

Applications of Linear Equations

Linear equations are used to model real-world situations, such as mixtures, pricing, and cost analysis.

Mixing Two Quantities

Use the standard equation where and are quantities, and are prices, and is the total value.

Example: Roses cost $3 each, total cost $36$:

• Equation:

• If (roses), (tulips)

Rate & Fixed Amount

Use the slope-intercept equation , where is the rate, is the fixed amount, and is the quantity.

Example: Metro Mobile charges per GB plus monthly fee:

• Equation:

• If GB,

Two Points of Information

Given two data points, calculate the slope and y-intercept, then write the equation.

Example: 20oz ketchup for , 36oz for :

• Slope:

• Using (20, 2.49):

• Equation:

• Price for 48oz:

Recap

• The y-value on the x-axis is always 0.

• The x-value on the y-axis is always 0.

• Standard Equation:

• Slope-Intercept Equation:

• Slopes of parallel lines are equal.

• Slopes of perpendicular lines are negative reciprocals.

Practice Exercises

• Section 1.3: 49, 53, 55, 59, 63, 65, 67, 69, 73, 74, 75

• Section 1.4: 7, 9, 11, 13, 15, 19, 21, 23, 31, 35, 37, 39, 43