# Point Slope Form of a Line

Examples, Graphs and Video Tutorial

#### 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?

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.

### Practice Problems

##### Problem 1

Identify which equations below 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 below 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').

##### Example 2

Find equation given 2 points.

Write the point slope equation of a line that goes through the points (1, 7) and (5, 19).

Step 1

Calculate the slope.

m (slope) =

Step 2

Substitute slope for 'm' and the coordinates for x1 and y1 into the formula (you can use either point).

• y − y1= m(x − x1)
• y − 7 = 3(x − 1)

Or, you can use the 2nd point.

• y − 5 = 3(x − 19)

### Practice writing equations in point slope form

##### Problem 1

This problem is just like example 1 (find equation given 1 point and the slope).

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

• y − y1 = 5(x − x1)
• y − 4 = 3(x − 1)

Write the point slope equation for the line in the graph below.

Step 1

Pick any 2 points on the line and calculate the slope.

Step 2

Substitute slope for 'm' and the coordinates for x1 and y1 into the formula (you can use either point).

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

Or if you use the 2nd point

• y − 0= − (x − 6)
• y = (x − 6)
##### Problem 2

This problem is just like example 1 (find equation given 1 point and the slope).

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

• y − y1 = -3(x − x1)
• y − 7 = -3(x + 4)
##### Problem 3

This problem is just like example 2 (find equation through 2 points).

Step 1

Calculate the slope.

m (slope) =

Step 2

Substitute slope for 'm' and the coordinates for x1 and y1 into the formula (you can use either point).

• y − y1 = m(x − x1)
• y − 6 = ½(x − 2)

Or, you can use the 2nd point.

• y − 7 = 3(x − 4)
##### Problem 4

This problem is just like example 1 (find equation given 1 point and the slope).

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

• y − y1 = ½ (x − x1)
• y − -4 = 3(x − 3)
• y + 4 = 3(x − 3)
##### Problem 5

This problem is just like example 2 (find equation through 2 points).

Step 1

Calculate the slope.

Step 2

Substitute slope for 'm' and the coordinates for x1 and y1 into the formula (you can use either point).

• y − y1= m(x − x1)

Or if you use the 2nd point.

#### Other Types of Problems

There are a many other question types that involve point slope form. Try some of the ones below to see how you do.

Write the point slope equation for the line in the graph below.

Step 1

Pick any 2 points on the line and calculate the slope

Step 2

Substitute slope for 'm' and the coordinates for x1 and y1 into the formula (you can use either point).

• y − y1= m(x − x1)
• y − 1 = − ½(x − 3)

Or if you can use the 2nd point.

• y − 01 = − ½(x − 5)
• y = − ½(x − 5)