Slope intercept form
Slopeintercept form one of a number of different forms of a linear equation. It is one of the most commonly used forms, and has the following structure:
y = mx + b
In the equation above, m is the slope, and b is the yintercept. Slope intercept form is useful because it allows us to quickly identify the slope and yintercept of a line, which in turn allows us to graph the line fairly easily.
Given a point and the slope of a line, you can write a linear equation in slopeintercept form.
Example
Given m = 3 and the point (3, 5), find b, and write the equation of the line in slopeintercept form.


Because we are given slope and a point on a line, we just need to find the yintercept in order to write the equation of the line in slopeintercept form. We found the yintercept by plugging the given values into the equation y = mx + b, then solving for b. Once the slope and yintercept are known, writing the equation of the line just involves plugging the slope and yintercept into m and b respectively.
How to find the slope and the yintercept
Given at least two points on a line, the slope of the line can be found using the slope formula:
For example, given the that (1, 5) and (2, 7) are points on the same line, the slope of the line can be found as follows:
The yintercept can be found in a number of ways. If a graph is given, the point at which the line intersects the yaxis is the yintercept. This occurs when x = 0. It follows that, given an equation, setting x equal to 0 and solving for y will yield the yintercept of the line. For example, given the line 3x + 2y = 6, the yintercept can be found by plugging 0 in for x, then solving for y.
3(0) + 2y = 6
2y = 6
y = 3
Thus, the yintercept occurs at the point (0, 3).
Other forms of linear equations
Slopeintercept form is just one of a number of different forms of linear equations, albeit it is the most commonly used form. Depending on the context that a linear equation is being used in, certain forms can be more beneficial to use. Other commonly used forms include pointslope form and standard form.
Slopeintercept form is useful because the slope and yintercept of the line can easily be read off the equation, which also makes it relatively easy to graph the line, since we can just start from the yintercept and find another point on the line by counting the rise over run (slope) in the coordinate plane.
Pointslope form
Pointslope form can be generalized as:
y  y_{1} = m(x  x_{1})
In the equation, y_{1} and x_{1} indicate a point on the line that is not the yintercept, while m is the slope. Pointslope form is useful when a point on the line and the slope of the line are known, since it enables us to write an equation for the line.
Standard form
The standard form of the equation of a line is
Ax + By = C
where A, B, and C are integers. Standard form is useful when we are trying to solve systems of linear equations. It can also be used to write the equation of vertical lines, something that cannot be done using slopeintercept or pointslope form.
Converting between different forms
It is relatively common to need to convert the equation of a line in standard or pointslope form to slopeintercept form.
Standard to slopeintercept form
Converting from standard form to slopeintercept form involves manipulating an equation in standard form so as to isolate y on one side of the equation such that the coefficient on the y is 1.
Example
Convert 2x  5y + 6 = 0 to slopeintercept form.
5y = 2x + 6
y = 2x/5 + 6/5
The slope is therefore 2/5, and the yintercept is at (0, 6/5).
Pointslope to slopeintercept form
Converting from pointslope form to slopeintercept form is relatively simple since both already include a slope, so all we need to do is convert the equation such that we can read off the yintercept, rather than some other point on the line.
Example
Convert y  7 = 1/2(x  4) to slopeintercept form.
y = 1/2x  4/2 + 7
y = 1/2x + 5