Equation of a line in point-slope form
What information do you need in order to find point-slope form of a line?
You have two options for writing the equation of a line: point-slope form and slope-intercept form. Both of them require that you know at least two of the following pieces of information about the line:
A point
Another point
The slope, ???m???
The ???y???-intercept, ???b??? (the ???y???-coordinate of the point at which the graph of the line crosses the ???y???-axis)
If you know any two of these things, you can find the equation of the line.
Point-slope form
The equation of a line in point-slope form can be written as
???y-y_1=m(x-x_1)???
In this form, ???(x_1,y_1)??? is a point on the line, and ???m??? is the slope. To use this form when you know two points on the line but you don’t know the slope, first find ???m??? using
???m=\frac{y_2-y_1}{x_2-x_1}???
Then simply plug the slope ???m??? and the coordinates of one point ???(x_1,y_1)??? into the point-slope form of the equation of a line.
How to find the equation of a line in point-slope form
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Examples of how to find point-slope form for the equation of a line when we know different pieces of information
Example
Write the equation of the line in point-slope form.
???m=-\frac{1}{4}???
???(-6,1)???
Since we’ve been given the slope of the line and a point on the line, we can use the point-slope form to find the equation of the line. We’ll plug ???m=-1/4??? and the coordinates of the point ???(-6,1)??? into the point-slope form of the equation of a line.
???y-y_1=m(x-x_1)???
???y-1=-\frac{1}{4}(x-(-6))???
???y-1=-\frac{1}{4}(x+6)???
Let’s try an example where we know two points on the line.
Example
Find the point-slope form of the equation of the line that passes through the points ???(-2,-4)??? and ???(3, 5)???. Use ???(-2,-4)??? for ???(x_1,y_1)???.
First, we need to find the slope of the line. It’s best to label the points before we plug them into the slope formula. We’ll say
???(-2,-4)=(x_1,y_1)???
???(3,5)=(x_2,y_2)???
Plug these into the formula for the slope.
???m=\frac{y_2-y_1}{x_2-x_1}???
???m=\frac{5-(-4)}{3-(-2)}???
???m=\frac{9}{5}???
Next, substitute ???m=9/5??? and the coordinates of the point ???(-2,-4)??? into the equation ???y-y_1=m(x-x_1)???. If you know two or more points on the line, as we do in this problem, you can use the coordinates of any point on the line, and you’ll get a correct equation for the line.
???y-(-4)=\frac{9}{5}(x-(-2))???
???y+4=\frac{9}{5}(x+2)???
Let’s try another example where we know two points on the line and need to find the equation of the line in point-slope form.
We’ll do one more.
Example
Find the point-slope form of the equation of the line that passes through the points ???(4,2)??? and ???(6,3)???.
We start by finding the slope.
???m=\frac{3-2}{6-4}=\frac{1}{2}???
Now plug in the slope and the coordinates of one of the points into the point-slope form of the equation of a line. We’ll use the point ???(4,2)???.
Even though we could simplify this further (by distributing the ???1/2??? over the two terms inside the parentheses), we’d end up with something that isn’t in point-slope form, so we leave it as is.