Math & History: Two Variable Linear Equations: Slope of a Line

I. Intro: The Slope Of A Line

The mathematical definition of slope is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines. By just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:

First, let’s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.

Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run.

II. Finding Slope From a Graph
As we saw in the previous section, the rate of change determines whether a set of data is linear or non-linear. In this section, we formalize how to find this rate of change.

The rate of change of or the distance between the y-values compared to the x-values is called the slope. The symbol for slope is 𝑚. As we will see, graphically, the slope measures how steep the line is.

One way to determine the slope is to plot the points on a graph, draw a line through them, and then count the change in the y-values, or rise, and the change in the x-values, or run.

Rise - Vertical change between two points.

Run - Horizontal change between two points.

The steps to finding the slope from a graph are:

1) Select any two random points on the graph of the line (preferably with integer coordinates).
2) Label them as A and B (in any order).

3) Calculate the "rise" from A to B. While going vertically from A to B, if we have to go
"up", then the rise is positive;
"down", then the rise is negative.
4) Calculate the "run" from A to B. While going horizontally from A to B, if we have to go
"right", then the run is positive;
"left", then the run is negative.
5) Now, use the formula: slope = rise/run

Example 1

Example 2

Example 3

Khan: Intro to Slope

III. Finding The Slope From Two Points

You can find the slope of the line by counting the rise over run. As we've alluded to above, you can also find the slope of a straight line without its graph if you know the coordinates of any two points on that line. Every point has a set of coordinates: an x-value and a y-value, written as an ordered pair (x, y). The x value tells you where a point is horizontally. The y value tells you where the point is vertically.

Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates (x₁,y₁) and Point 2 has coordinates (x₂,y₂).

The rise is the vertical distance between the two points, which is the difference between their y-coordinates. That makes the rise (y₂−y₁). The run between these two points is the difference in the x-coordinates, or (x₂−x₁).

An alternate way to present the formula is:

Slope = Δy/Δx

The little triangle symbol is the Greek uppercase letter delta which means "change" or "difference in".

Example

In the example you’ll see that the line has two points each indicated as an ordered pair. The point (0,2) is indicated as Point 1, and (−2,6) as Point 2. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run.

You can see from the graph that the rise going from Point 1 to Point 2 is 4, because you are moving 4 units in a positive direction (up). The run is −2−2, because you are then moving in a negative direction (left) 2 units. Using the slope formula,

Slope = rise/run = 4/-2 = -2

You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Let’s organize the information about the two points:

Can Either Point Be (X₁, Y₁)?

Let's find out.

IV: Positive & Negative Slopes

When we look at a line on a graph, its slope can fall into one of four categories. Each type of slope tells us something different about how the line behaves:

1. Positive Slope: A line with a positive slope rises as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values also increase. Think of it as going uphill.

2. Negative Slope: A line with a negative slope falls as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values decrease. Think of it as going downhill.

3. Zero Slope: A flat, horizontal line has a zero slope. It means there’s no change in y-values as the x-values increase. The line stays level.

4. Undefined Slope: A vertical line has an undefined slope. This is because the x-values don’t change, while the y-values might change infinitely.

Khan: Positive & Negative Slope

V. Slope-Intercept Form

There are many ways you can write a linear equation. For instance:

-2x + y =3

could be rewritten as

y - 5 = 2(x - 1)

or rewritten as

y = 2x + 3

This last form is referred to as slope-intercept form.

A. What is slope-intercept form?

Slope-intercept is a specific form of linear equations. It has the following general structure.

y = mx + b

Here, m and b can be any two real numbers.

B. The coefficient & constant in slope-intercept form

Besides being neat and simplified, slope-intercept form's advantage is that it gives two main features of the line it represents:

The slope is m
The y-coordinate of the y-intercept is b. In other words, the line's y-intercept is at (0, b).

For example, the line y = 2x + 1 has a slope of 2 and a y-intercept at (0,1):

The fact that this form gives the slope and the y-intercept is the reason why it is called slope-intercept in the first place!

C. Graphing Lines From Slope-Intercept Form

Graph y = 2x + 7

Straight from the equation, we know the y-intercept is 7 and we know the slope is 2.

So we first mark the y-intercept on the graph (0,7) then for every one unit we go to the right, we must go up two units:

D. Writing Slope-Intercept Equations

1. Writing equations from y-intercept and another point

Let's write the equation of the line that passes through the points (0,3) and (2,7) in slope-intercept form.

Recall that in the general slope-intercept equation y = mx + b, the slope is given by m and the y-intercept is given by b.

Finding b

The y-intercept of the line is (0,3), so we know that b = 3.

Finding m

Recall that the slope of a line is the ratio of the change in y over the change in x between any two points on the line: