Finding Slope from Two Points in Geometry: A Step-by-Step Guide

What is Slope?

Slope can be calculated by finding the rise (vertical change) over the run (horizontal change) between two points. The formula for finding slope is:

Slope = (change in y-value) � (change in x-value)

Finding Slope from Two Points

To find the slope from two points, we need to first find the coordinates of both points. The coordinates of a point are expressed as (x, y).

Once we have the coordinates, we can find the change in x-value and y-value between the two points. This is done by subtracting the x-value and y-value of the first point from the x-value and y-value of the second point respectively.

We can then plug those values into the slope formula to calculate the slope between the two points.

Example Problem

Let's look at a specific example. Suppose we have two points, A and B, located at (2, 3) and (7, 8) respectively.

First, we need to find the change in x-value and y-value between the two points. This is done by subtracting the x-value and y-value of the first point from the x-value and y-value of the second point.

So, the change in x-value is (7 - 2) = 5, and the change in y-value is (8 - 3) = 5.

We can then plug those values into the slope formula to calculate the slope between the two points:

Slope = (change in y-value) � (change in x-value) = 5 � 5 = 1

Therefore, the slope between points A and B is 1.

Graphing Slope

We can also use the slope to graph a line that passes through both points. To do this, we need to find the y-intercept of the line first. This can be done by substituting the x-value and y-value of the first point into the slope formula and solving for b, the y-intercept.

In our example, the first point is (2, 3). So, we can substitute 2 for x and 3 for y in the slope formula and solve for b.

m = (y2 - y1) � (x2 - x1) = (3 - b) � (2 - 0) = 1

We can then solve for b:

1 = (3 - b) � 2

2 = 3 - b

2 + b = 3

b = 1

Therefore, the y-intercept of the line passing through points A and B is 1.

We can then use the slope and y-intercept to graph the line. The slope is 1, so we can start at the y-intercept of 1 and draw a line that rises 1 unit for every unit it runs.

Practice Problems

Try these practice problems to test your understanding of finding slope from two points.

1. Find the slope between points (3, 4) and (10, 8).
2. Find the y-intercept of the line passing through points (4, 5) and (7, 9).
3. Find the slope between points (2, 5) and (2, 9).
4. Find the y-intercept of the line passing through points (5, 7) and (10, 12).
5. Find the slope between points (2, 1) and (6, 6).

Answers

1. Slope = (8 - 4) � (10 - 3) = 4 � 7 = 0.571
2. y-intercept = (5 - (7/4)) � (9 - 5) = -2/2 = -1
3. Slope = (9 - 5) � (2 - 2) = undefined
4. y-intercept = (7 - (10/5)) � (12 - 7) = -3/5 = -0.6
5. Slope = (6 - 1) � (6 - 2) = 5 � 4 = 1.25

Summary

In this lesson, we learned how to find the slope between two points given their x- and y-coordinates. We also learned how to use the slope to graph a line that passes through the two points, as well as how to calculate the y-intercept of the line. With these skills, we can easily find the slope of any two points and graph the line they form.

FAQ

What is the formula for slope?

The formula for slope is m = (y2 - y1)/(x2 - x1).

When given two points, how do you find the slope?

When given two points, you can find the slope by plugging in the coordinates of the points into the formula for slope. For example, if you have two points (x1, y1) and (x2, y2), you would plug in the coordinates for x1, y1, x2, and y2 into the slope formula, and you would calculate m = (y2 - y1)/(x2 - x1).