Introduction

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

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:   

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

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.

 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.

Step 1: Find the Slope

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

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.

Step 1: Find the Slope

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

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

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

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 =x + 1 

2)  y = -x + 6

3)  y = ½ x – 3

4)  y = 3  (or y = 0x + 3)

5) y =