Point Slope Form of a Line, examples, graphs and video tutorial and many practice problems with step by step work shown

Overview of different forms of a line's equation

There are many different ways that you can express the equation of a line. There is the slope intercept form, standard form and also this page's topic - point slope form. Each one expresses the equation of a line, and each one has its own pros and cons. Point slope form, this page's topic, makes it easy to find the line's equation when you only know the slope and a single point on the line (see example 1). Point slope form is also the quickest method for finding the equation of line given two points (see example 2).

Definition of the Point Slope Form Equation

$ \text{Point Slope Formula} \\ y-y_1 = m (x - x_1) $

Examples of Point Slope Form Equations	Non-Examples
y − 5 = 3(x − 2)	2y = 4x + 2
y + 5 = ½(x − 12)	x = 6 − y
y + 5 = 4(x + 12)	y = 2x + 7
y + ½= ¼(x + ¾)	3x + 5y = 2

When is point slope form useful?

When you do not know the intercepts (x and y-intercepts ). (Instead, use Standard Form in this case)
When you do know or can easily get the slope and 1 point.

For example, if you know that the slope of a particular line is $$5$$ and that that this line goes through the point $$ (1,5)$$.

As long as you are not concerned with the intercepts, then it would be very easy to express the equation of this example line in point slope form.

When are other forms more useful?

Standard Form is more convenient when you know or need to know the x-intercept and also the y-intercept and do not know the slope.
Slope intercept form is best when you know the slope and the y-intercept.

Video Tutorial on Point Slope Form

Practice Problems

Problem 1

Identify which equations belowon the right are in point slope form?

Equation 1: 2x + 2y = 6
Equation 2: y - 5 = 3(x -2)
Equation 3: y = 2x + 3
Equation 4: y + 3 = 3(x - ½)

Equation 2 and equation 4 are the only ones in point slope form.

Equation 1 is in Standard Form

Equation 3 is in Slope intercept form

Problem 2

Which equations belowon the right are in point slope form?

Equation 1: y − 2 = 3(x − 4)
Equation 2: y + 5 = (x − 4)
Equation 3: y = 3(x − 4) + 2
Equation 4: y − 2 = 3(x − 4)

Equation 1, 2 and equation 4 are the in point slope.

How to Solve common point slope form questions

Example 1

Write the point slope equation of a line with slope 3 that passes through the point (-2, 5)

Step 1

Substitute slope for 'm' and the coordinates for x₁ and y₁ into the formula.

y − y₁ = m(x − x₁)
y − 5 = 3(x − -2)
y − 5 = 3(x + 2)

Yes, it really is that easy to write the equation of a line in point slope form when you know its slope and 1 point on the line! This should be a real relief to those of you who are used to doing this with slope intercept form, which would require 2 more steps (i.e. substitute a point and solve for b').