←
Writing an Equation Given Two Points

1.12.5 Writing an Equation Given Two Points

1.12.5 • Writing an Equation Given Two Points

Activity

Answer the following questions.

1.

Use the graph below to pick two points and find the slope between the two points.

A graph with an orange downward-sloping line passing through the points (-2, 3) and (4, 1), marked by blue dots. The x-axis ranges from -8 to 8, and the y-axis ranges from 0 to 6.

Compare your answer:

The slope of the line is −13−13.

2.

Use your two points in the slope formula to confirm the slope is correct.

Compare your answer:

Your answer may vary, but here is a sample.

m=y2−y1x2−x1=1−34−(−2)=−26=−13m=y2−y1x2−x1=1−34−(−2)=−26=−13

3.

Using the slope and a point, write your equation in point-slope form.

Compare your answer:

Your answer may vary, but here is a sample.

m=−13m=−13 and (4,1)(4,1)

y−y1=m(x−x1)y−y1=m(x−x1) becomes: y−1=−13(x−4)y−1=−13(x−4).

4.

Now write your equation in slope-intercept form.

Compare your answer:

y=−13x+73y=−13x+73

5.

Convert slope-intercept form to standard form.

Compare your answer:

(x+3y)=7(x+3y)=7

6.

Look at your equation and the original graph. What do you notice? How can you use a graph to check your equation?

Compare your answer:

Your answer may vary, but here is a sample.

The yy-intercept of the graph is 7373 and the slope is −13−13.

Self Check

Which equation represents the slope-intercept form of the equation of a line containing the points  ( − 2 , − 4 ) and ( 1 , − 3 ) ?
  1. y = 3 x − 6
  2. y = 7 x + 10
  3. y = 1 3 x − 7 3
  4. y = 1 3 x − 10 3

Additional Resources

Find an Equation of the Line Given Two Points

So far, you have two options for finding an equation of a line: slope-intercept or point-slope. When you start with two points, it makes more sense to use the point-slope form.

But then you need the slope. You can find the slope with just two points and then use it and one of the given points to find the equation.

Example

Write the equation of a line that contains the points (−3,−1)(−3,−1) and (2,−2)(2,−2). Write the equation in slope-intercept form and standard form.

Step 1 - Find the slope using the given points.

Find the slope of the line through (−3,−1)(−3,−1) and (2,−2)(2,−2).

m=y2−y1x2−x1m=y2−y1x2−x1

m=−2−(−1)2−(−3)m=−2−(−1)2−(−3)

m=−15m=−15

m=−15m=−15

Step 2 - Choose one point.

Choose either point.

(x1,y1)(x1,y1)

(2,−2)(2,−2)

Step 3 - Substitute the values into the point-slope form, y−y1=m(x−x1)y−y1=m(x−x1).

Simplify

y−y1=m(x−x1)y−(−2)=−15(x−2)y+2=−15x+25y−y1=m(x−x1)y−(−2)=−15(x−2)y+2=−15x+25

Step 4 - Write the equation in slope-intercept form.

y=−15x−85y=−15x−85

Step 5 - Convert slope-intercept form to standard form.

y=−15x−85y=−15x−85

15x+y=−8515x+y=−85

5(15x+y)=5(−85)5(15x+y)=5(−85)

(1x+5y)=−8(1x+5y)=−8

Use this table for easy reference to find an equation of a line given two points.

Step 1 - Find the slope using the given points.

m=y2−y1x2−x1m=y2−y1x2−x1

Step 2 - Choose one point.

Step 3 - Substitute the values into the point-slope form: y−y1=m(x−x1)y−y1=m(x−x1).

Step 4 - Write the equation in slope-intercept form.

Step 5 - Convert slope-intercept form to standard form.

To find an equation of a line given the slope and a point, follow these steps:

Step 1 - Identify the slope.

Step 2 - Identify the point.

Step 3 - Substitute the values into the point-slope form, y−y1=m(x−x1)y−y1=m(x−x1).

Step 4 - Write the equation in slope-intercept form.

Step 5 - Convert slope-intercept form to standard form.

Try it

Writing the Equation of a Line Given Two Points

Write the equation of a line that contains the points (−3,5)(−3,5) and (−3,4)(−3,4).

Here is how to find an equation of a line that contains the points (−3,5)(−3,5) and (−3,4)(−3,4).

Again, the first step will be to find the slope.

Step 1 - Find the slope of the line through (−3,5)(−3,5) and (−3,4)(−3,4).

Step 2 - m=y2−y1x2−x1m=y2−y1x2−x1

Step 3 - m=4−5−3−(−3)m=4−5−3−(−3)

Step 4 - m=−10m=−10

A line with undefined slope is a vertical line. Both given points have an xx-coordinate of −3−3. So, the equation of the line is x=−3x=−3. There is no yy, so the equation cannot be written in slope-intercept form.