Writing Equations
Slope-Intercept Form
There are several forms that the equation of a line can take. They may look different, but they all describe the same line--a line can be described by many equations. All (linear) equations describing a particular line, however, are equivalent.
The first of the forms for a linear equation is slope-intercept form. Equations in slope-intercept form look like this:
| y = mx + b |
where m is the slope of the line and b is the y-intercept of the line, or the y-coordinate of the point at which the line crosses the y-axis.
To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope. This is the value of m in the equation. Next, find the coordinates of the y -intercept--this should be of the form (0, b) . The y - coordinate is the value of b in the equation.
Finally, write the equation, substituting numerical values in for m and b . Check your equation by picking a point on the line (not the y -intercept) and plugging it in to see if it satisfies the equation.
Example 1: Write an equation of the following line in slope-intercept form:
First, pick two points on the line--for example, (2, 1) and (4, 0) . Use these points to calculate the slope: m =
= 
= - 
.Next, find the y -intercept: (0, 2) . Thus, b = 2 .
Therefore, the equation for this line is y = -
x + 2
.Check using the point (4, 0) : 0 = -
(4) + 2
? Yes.
Example 2: Write an equation of the line with slope
m = 
which crosses the
y
-axis at
(0, - 
)
.
y =
x -
Example 3: Write an equation of the line with
y
-intercept 3 that is parallel to the line
y = 7x - 9
.
Since
y = 7x - 9
is in slope-intercept form, its slope is
7
.
Since parallel lines have the same slope, the slope of the new line will also be
7
.
m = 7
.
b = 3
.
Thus, the equation of the line is
y = 7x + 3
.
Example 4: Write an equation of the line with
y
-intercept
4
that is perpendicular to the line
3y - x = 9
.
The slope of
3y - x = 9
is
.
Since the slopes of perpendicular lines are opposite reciprocals,
m = - 3
.
b = 4
.
Thus, the equation of the line is
y = - 3x + 4
.
