Hi there. In this math education post, I cover the point slope form of the line. This topic is for early high school mathematics students. (It is assumed that the reader is familiar with the distributive property for the second section of this post.)
Given a single point (a, b) on the line and a slope m of that line, the point-slope form of the line that passes through (a, b) with that slope can be written as:
The x-coordinate of the point (a, b) which is a is associated with the x in x - a. The y co-ordinate from the point (a, b) which is b is associated with y in y - b.
The slope of the line m is on the right side in front of (x - a).
Example One
Suppose you are given the point (1, 7) with a slope of 3. The point slope form of this line that passes through the point (1, 7) with a slope of 3 is:
Example Two - Negative Coordinates
This example features negative numbers which is sometimes challenging for those starting out with algebra. Getting negative numbers wrong is a common mistake in early high school mathematics.
You are given the point (-3, - 2) with a slope of -8. What is the point-slope form of this line passing through (-3, -2) and slope -8?
Answer
Keep in mind that you would negative of a negative which turns into a positive. On the left side y minus negative 3 turns into y + 3. The right side has x - (-2) which turns into x + 2.
The point-slope form for this example turns out to be the following.
y = mx + b Form For A LineThe point slope form is a nice equation form for a line as you need a point and a slope. One problem is that the y-intercept is not easy to pick out from the point slope form. You would need the y = mx + b form of the line in order to obtain the y-intercept.
Example One
The point-slope form of a line that goes through the point (1, 1) with a slope of 2 is:
To obtain the y = mx + b form of the line, simply use algebra and distributive law to isolate for the y variable.
Yes the y variable is isolated but it is not in the y= mx + b form. Apply the distributive law and simplify.
Example Two
Given a point of (-1, -3) and a slope of -5 for a line, what is the y = mx + b form for this line?
Start with the point-slope form.
Move the 3 from the left to the right.
Apply distributive law and simplify.
Example Three - General Case
I was thinking of doing the general case first but I thought it would be a bit too much early on. Given the general formula for the point slope, you can convert this into the y = mx + b form for the equation of a line.
Start with the point-slope form of the line passing through the point (a, b) with a slope m.
Add the value a to both sides which isolates for y.
Apply distributive law to fully simplify.
Note that the slope is the m in front of the variable x. The y-intercept is now -mb + a.
Thank you for reading.