Point slope versus slope intercept of a line (Algebra 1)

As one of the first exercises to writing the equation of a line for the Algebra 1 student, they are always introduced to the concept of the slope intercept form of the line.  As they get more proficient with the slope intercept form of the line, they will then be asked to use the point slope form of the same line.  In this article, we will discuss and show that the slope intercept form and the point slope form of an equation is exactly the same as the other.

Slope intercept form of a line is

y = mx + b

(x, y) is the coordinate of any point on the line.
m is the slope of the line.
b is the y-intercept of the line (y-intercept means that the point where the line crossed the y axis)

Whereas point slope form of a line is

(y - y1) = m(x - x1)

(x1,  y1) is the coordinate of any point on the line.
m is the slope of the line.

Students should note that there is ABSOLUTELY NO DIFFERENCE in the final outcome of the equation of a line if it is written in the point slope form, or the slope intercept form.  Once you have written the equation in one of the format (whether it be point slope form, or slope intercept form), you can translate this equation to the other form just as easily. 

The reason why that the line of an equation describes the characteristics of the line, so no matter what format you decide to write the equation in, it is still describing the same line. 

Let us use a concrete example to prove that point slope form of an equation is exactly the same as the slope intercept form of the equation.  You are given that the slope of a line is 4, and this line contains a point (-1, -6).  Let's work on the slope intercept form of the equation first.

y = mx + b
-6 = 4(-1) + b
-6 = -4 + b
b = -2

y = 4x -2 -> (slope intercept form)

To write this equation in the point slope form, we will use the equation of

(y - y1) = m(x - x1)
(y + 6) = 4(x + 1) -> (point slope form)

Let's prove that the point slope form of an equation is exactly the same as the slope intercept version.
  
y + 6 = 4x + 4 -> taken from point slope form
y = 4x - 2

As you can see, it is easy to derive one form of the equation to another.  Here are some more problems that you should use to practice on writing the equation of a line using both slope intercept form and point slope form.

1. You are given 2 points (-2, 4) and (1, 2)
   
2. m = -3, (4,3)