Log in Register

Login to your account

Username *
Password *
Remember Me

Create an account

Fields marked with an asterisk (*) are required.
Name *
Username *
Password *
Verify password *
Email *
Verify email *

Newsletter

Please select a list in the module configuration!No fields selected! Select at least "email"!

Find the equation of the line (given 2 points)


Introduction

Find_the_eqn_given_2_points_vis_1

One of the basic truths of geometry is that there is only one way to connect any two points with a straight line.  On the coordinate plane, any two points can be connected by a single line that has a single equation.  This lesson describes how to find the equation of a line that goes through any two points.

Lesson

Find_the_eqn_given_2_points_vis_2a

 

The line that connects two points on the coordinate plane can easily be drawn on paper with a ruler.  Move your cursor over the coordinate plane to the right to see the line that connects the two points.

 

To find the equation of the line, we must first find the slope.  To do so, use the slope ratio:

 

Slope ratio:  Find_the_eqn_given_2_points_vis_3 

We now know that the line goes through the points (2, 3) and (8, 6) and has a slope ofFind_the_eqn_given_2_points_vis_4

 

In order to find the equation of the line, we actually only need one coordinate as well as the slope.  Since there are two known coordinates, simply pick one and use it in the formula.

 

Remember that the point-slope form of the graph is y – y1 = m(x – x1), where the coordinate of the point is (x1, y1) and the slope is m.

Find_the_eqn_given_2_points_vis_5 

 

 You will have a choice of two coordinates when finding the equation in this lesson.  Before choosing one, take a closer look at the equation and pick the coordinate that looks the easiest to put into the equation.  Usually, this means picking points that have  smaller coordinates (in the first quadrant where both coordinates are positive).  Example 1 contains one coordinate that is in the first quadrant and the problem is a little bit easier when picking this coordinate.

 

Example 1:  Find the equation of the line that contains the points (4, 1) and (-2, -17).  Then change the equation into slope-intercept form.

 

Find_the_eqn_given_2_points_vis_8aStep 1: Find the Slope

Find_the_eqn_given_2_points_vis_6

Step 2:  Pick a point to use in the equation

    (4, 1) has two smaller positive numbers, so pick it

 

Step 3:  Find the equation and change to slope-intercept form

 

Find_the_eqn_given_2_points_vis_7 

 

Step 4:  Double check your answer to see if it makes sense.  Scroll over the graph to the right and see if a slope of 3 and y-intercept of -11 are reasonable answers for the line.

 

Example 2 shows contains two points that both have the same y-coordinate.

 

Example 2:  Find the equation of the line that contains the points (6, -2) and (-3, -2).  Then change the equation into slope-intercept form.

 

Find_the_eqn_given_2_points_vis_9a

 

Step 1: Find the Slope

Find_the_eqn_given_2_points_vis_10 

 

Step 2:  Pick a point to use in the equation

 

           Either point is fine.  (6, -2) was picked here.

 

Step 3:  Find the equation and change to slope-intercept form

 

Find_the_eqn_given_2_points_vis_11

 

Step 4:  Double check your answer to see if it makes sense.  Scroll over the graph to the right and see if a slope of 0 and y-intercept of -2 are reasonable answers for the line.

 

 

You may not have time to draw the graph for every problem you do.  If not, be very careful when finding the slope and simplifying the equation.

 

Example 3:  Find the equation of the line that contains the points (-3, 12) and (21, -4).  Then change the equation into slope-intercept form.

 

Step 1: Find the Slope

 

Find_the_eqn_given_2_points_vis_12 

 

Step 2:  Pick a point to use in the equation 

          Either point is fine.  (6, -2) was picked here.

Step 3:  Find the equation and change to slope-intercept form

 

Find_the_eqn_given_2_points_vis_13 

 

 

 

Take your time and show all your work and you can be successful in finding equations of lines through two points.  Resist the temptation to do too much in your head to avoid mistakes with the calculations involved here.

 

Try It

 

 

Find slope-intercept equation of a line containing: 

1)  Points (3, 3) and (7, 9)

2)  Points (5, 1) and (1, 5) 

3)  Points (-2, -4) and (6, 0)

4)  Points (0, 3) and (8, 3) 

5)  Points (0, 0) and (21,15)

 

Scroll down for solutions...

 

 

 

 

 

 

 

 

 

Solutions:

1)  y =Find_the_eqn_given_2_points_vis_14x + 1 

 

 

2)  y = -x + 6

3)  y = ½ x – 3

4)  y = 3  (or y = 0x + 3)

 

5) y = Find_the_eqn_given_2_points_vis_15

 

Related Links:

For more information on slope-intercept form and related topics, try one of the links below.


Resource Pages

 

Related Lessons

 

Looking for something else?  Try the general math or algebra lessons.

Copyright © 2014. Free Math Resource.
All Rights Reserved.