16 Finding a Linear Equation
Perhaps the most familiar form of a linear equation is the slope-intercept form, written as
The Slope of a Line
The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.
If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in Figure 2. The lines indicate the following slopes:

A General Note: The Slope of a Line
The slope of a line, m, represents the change in y over the change in x. Given two points,
Example 7: Finding the Slope of a Line Given Two Points
Find the slope of a line that passes through the points
Solution
We substitute the y-values and the x-values into the formula.
The slope is
Analysis of the Solution
It does not matter which point is called
Example 8: Identifying the Slope and y-intercept of a Line Given an Equation
Identify the slope and y-intercept, given the equation
Solution
As the line is in
Analysis of the Solution
The y-intercept is the point at which the line crosses the y-axis. On the y-axis,
The Point-Slope Formula
Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.
This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.
A General Note: The Point-Slope Formula
Given one point and the slope, the point-slope formula will lead to the equation of a line:
Example 9: Finding the Equation of a Line Given the Slope and One Point
Write the equation of the line with slope
Solution
Using the point-slope formula, substitute
Analysis of the Solution
Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.
Try It 8
Given
Example 10: Finding the Equation of a Line Passing Through Two Given Points
Find the equation of the line passing through the points
Solution
First, we calculate the slope using the slope formula and two points.
Next, we use the point-slope formula with the slope of
In slope-intercept form, the equation is written as
Analysis of the Solution
To prove that either point can be used, let us use the second point
We see that the same line will be obtained using either point. This makes sense because we used both points to calculate the slope.
Standard Form of a Line
Another way that we can represent the equation of a line is in standard form. Standard form is given as
where
Example 11: Finding the Equation of a Line and Writing It in Standard Form
Find the equation of the line with
Solution
We begin using the point-slope formula.
From here, we multiply through by 2, as no fractions are permitted in standard form, and then move both variables to the left aside of the equal sign and move the constants to the right.
This equation is now written in standard form.
Try It 9
Find the equation of the line in standard form with slope
Vertical and Horizontal Lines
The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as
where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c.
Suppose that we want to find the equation of a line containing the following points:
Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through
The equation of a horizontal line is given as
where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c.
Suppose we want to find the equation of a line that contains the following set of points:
Use any point for
The graph is a horizontal line through

Example 12: Finding the Equation of a Line Passing Through the Given Points
Find the equation of the line passing through the given points:
Solution
The x-coordinate of both points is 1. Therefore, we have a vertical line,