BackLinear Equations, Graphs, and Functions: Writing Equations of Lines 2.3
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Linear Equations, Graphs, and Functions
Writing Equations of Lines
This topic covers the foundational concepts of linear equations, including how to write equations of lines given various information, graph lines, and understand the properties of horizontal and vertical lines. These skills are essential for analyzing and interpreting linear relationships in College Algebra.
Objective: Learn to write equations of lines given slope and y-intercept, graph lines using slope and y-intercept, write equations given slope and a point, write equations given two points, and write equations of horizontal and vertical lines.
Slope-Intercept Form: The equation of a line in slope-intercept form is , where m is the slope and b is the y-intercept.
Example: Write the equation of a line with slope and y-intercept :
Uniqueness of Linear Equations: Every nonvertical line has a unique equation in slope-intercept form.
Graphing Lines Using Slope and Y-Intercept
To graph a line using its slope and y-intercept, start by plotting the y-intercept, then use the slope to determine the rise and run to plot additional points.
Example: Graph - Slope (rise 3, run 1) - Y-intercept (plot point at (0, -6)) - From (0, -6), move up 3 units and right 1 unit to plot the next point.
Converting to Slope-Intercept Form: For equations not in slope-intercept form, solve for to rewrite as . Example: - Subtract : - Divide by 3:
Point-Slope Form of a Line
The point-slope form is useful when you know the slope and a point on the line. The general form is .
Example: Write the equation of a line with slope passing through : - Distribute: - Add 5:
Writing Equations Given Two Points
To write the equation of a line passing through two points, first calculate the slope, then use point-slope or slope-intercept form.
Slope Formula:
Example: Find the equation of the line passing through and : - Slope: - Use point-slope form: - Multiply both sides by 9 to clear fractions: - Rearranged: (Standard form)
Equations of Horizontal and Vertical Lines
Horizontal and vertical lines have special forms. Horizontal lines have a slope of zero, while vertical lines have an undefined slope.
Horizontal Line: Equation is , where is the y-value for all points on the line. Example: Line passing through :
Vertical Line: Equation is , where is the x-value for all points on the line. Example: Line passing through :
Summary Table: Forms of Linear Equations
Form | Equation | When to Use |
|---|---|---|
Slope-Intercept | Given slope and y-intercept | |
Point-Slope | Given slope and a point | |
Standard | General form, often for integer coefficients | |
Horizontal Line | All points have the same y-value | |
Vertical Line | All points have the same x-value |
Additional info:
Lines parallel to a given line have the same slope.
Lines perpendicular to a given line have slopes that are negative reciprocals.
Linear functions are defined by equations in slope-intercept form.
