Equation of a Line Given Slope and a Point

How to write the equation

How to write equation of a line, given the slope of a line and also one of points on the line.

Students are often asked to find the equation of a line that passes through a point and has a certain slope.. Watch the video tutorial below to understand how to do these problems and, if you want, download this free worksheet if you want some extra practice.

Video Tutorial on Equation from Slope and a Point

Steps to follow:
Example: How to find the equation of a line that goes through the point (1, 3) and has a slope of 2.
What we know from the problem: that the slope is 2 so we can write y = 2x + b
Our goal: to find the y-intercept or "b"

Step 1

Substitute slope for 'm' in y = mx + b
y = mx +b
y = 2x + b

Step 2

Substitute the point (1,3) into equation
y = 2x + b
3 = 2(1) +b

Step 3

Solve for b
3 = 2+b
3-2 = 1= b

Step 4

Now that we know b, all that we have to do is put it into the equation and we now have our line in slope intercept form
y = 2x+1

Below is a picture of y = 2x + 1


Equation of line from its slope and one point

Practice Problems

Try to write (or mentally make a note) of each equation below.

Problem 1

Directions: For all questions below, express your answers using slope intercept form.

Write the equation of a line that has a slope of ½ and that goes through the point (2,3)

Goal

Find "b", the y-intercept

y = ½x + b

Step 1

Substitute in the point (2,3)

3 = ½(2)+ b

Step 2

Solve equation for b

3 = 1 + b
3 - 1 = 2 = b

Last Step

Insert b into equation

y = ½x +2


Below is graph of y = ½x +2

Diagram of Line's Equation
Problem 2

The slope of a line is 10 and the line goes through the point (3, 34). Express the equation of this line in slope-intercept form.

Step 1

Plug slope into M

y = 10x + b

Step 2

Substitute x and y coordinates of point

34 = 10 •3 + b   28 = 30 + b

Step 3

Solve equation for B

34 - 30 = b
4 = b

Step 4

Plug in B as the y-intercept and write standard form equation.

y = 10x + 4

Problem 3

The slope of a line is 3 and the line goes through the point (6,23), express the equation of this line in slope-intercept form.

Step 1

Plug slope into M

y = 3x + b

Step 2

Substitute x and y coordinates of point

23 = 3 × 6 +b   23 = 18 + b

Step 3

Solve equation for B

23 - 18 = b = 5

Step 4

Plug in B as the y-intercept and write standard form equation.

y = 3x + 5

Problem 4

What is the equation of a line whose slope is 0 and that goes through the point (7,5)? Express the equation of this line in slope-intercept form.

There are two ways to approach this problem. First, you can look at this as a horizontal line, any horizontal line has the standard form equation of y = b where b is the y intercept. Since the slope is 0, and only horizontal lines have a slope of zero, all points on this line including the y-intercept must have the same y value. This y-value is 5, which we can get from the fact that the line passes through the point (7,5). Therefore the standard form equation of this horizontal line is y = 5.

Step 1

Plug slope into M

y = 0x + b

Step 2

Substitute x and y coordinates of point

5 = 0 • 7 + b   5 = b

Step 3

Solve equation for B

b = 5

Step 4

Plug in B as the y-intercept and write standard form equation.

y = 0x + 5 y = 5

Problem 5

If the slope of a line is undefined and line goes through the point (-3, -13), express the equation of this line in slope-intercept form.

Since the slope is undefined, the line is a vertical line. Therefore, the equation is just x = -3.


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