Slope
Slope is a key concept in mathematics, not just limited to linear equations! Slope shows the relationship between inputs and outputs, shows how a relationship changes. As such, this concept needs some special attention.
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Slope is said to be, “rise over run.” Rise is up (or down if it is negative), and run is horizontal. Slope is then, change in y over change in x.
These are great ways to think about finding slope, but doesn’t really explain what slope is. Slope is a way to describe the relationship between consecutive pairs of inputs and outputs. That’s why slope is often called, rate of change. Rate of change makes a lot of sense in real-world contexts. For example, if have a job where you make an hourly wage. That hourly wage is the output and the time you spend at work is the input. For every hour you work (increment of input), you make ____ dollars. The more hours you work the more money you make.
Other equation types have slopes, but they’re not constant. Only lines have a constant slope. There are three ways we will discuss to find slope. After those three ways, we will discuss the two strange lines, the horizontal and the vertical.
Graphs are read from left to right, just like
the English language. A line that goes down, when read from left to right, has a negative slope! That’s the first thing to determine when counting slope.
After you know the sign, find two “lattice point intersections.” That means, places where the graph crosses the coordinate plane at a whole-number coordinate. We start at lattice points because we know what the value at that point is. Everywhere else, we have to estimate.
Once you find two points on your line that lie on lattice points, count the change in y (rise) and the change in x (run). Be sure that you don’t mess up the units as labeled on each axis. Write your values in a fraction, y over x, and REDUCE. Do not write your answer as a mixed number or a decimal. They may be valid equivalent values, but the information they share is a little harder to decode than just rise/run.
In this diagram we can see that the slope is 
Do not write 
as you need to reduce. Also, do not write a slanted fraction bar, like 1/3. This will make the mathematical operations we must perform with slope difficult and confusing. You’ll too easily confuse what is a numerator and what is a denominator.
Method 2: Identifying from an Equation
The letter x is input, and y is output. For slope we use the letter m, as in y = mx + b. The coefficient of x, when written in this form, is the slope. (Form means exact pattern.) So, if you saw the equation y = -2x + 5, you would instantly recognize that -2 is the slope. That means that the graph is going down from left to right.
But, what if the equation was in Standard Form, like 2x – y = -5 (they’re actually the same equation). Remember, standard form of a linear equation is Ax + By = C. In this case you have two choices. You can solve for y, or you can take the ratio of 
Here, A = 2 and B = -2, so the ratio would be -2/1, which is just -2.
Method 3: The Formula
When you count the difference in y, you are actually subtracting. Suppose you had two points, the y – coordinate for the first was at 8, and the y – coordinate for the second was at 2. The difference is six, whether you count each incremental unit of change or subtract them all at once. That’s where we get the following formula.
Because subtracting the consecutive y – coordinates is how you find the difference is them, and y is “rise,” and the same works for the x – coordinates, which is run, and slope is “rise over run,” this formula is the same as counting, but not visual. You do not need a graph.
ADVICE: If you’re given two points and asked to find the slope, do not sketch a graph and try to count. Use the formula!
Common Pitfalls: It does not matter where you start counting on a graph, and it does not matter which point you call point 1 or point 2. But, whatever you pick to be the first y, it’s order pair matching x must be the first x.
The minus sign in the middle must remain. Use parenthesis when plugging in a negative number. It will keep you from making this mistake.
Example: Find the slope of the line containing the points (3, -2) and (-6, 7).
All we need to do is determine which point we’d like to be point 1, and which to be point 2. There is not a right or wrong choice here, just pick. The answer is the same either way.
This slope means that for an increment of 5 units of input, the output decreases by 13. If you wrote this as a mixed number,
that information gets lost.
Exceptions
The two exceptions for slope are the same to types of linear equations that cause us trouble with domain and range. They are the horizontal and vertical lines. Let’s talk about each briefly so you really understand what’s happening. That way you don’t have to rely on your memorization of facts, but instead understand and can better recall.
Horizontal Line
y = #
To the left we see the line y = -3. Every point on the line has a y – coordinate of -3.
Basic Meaning of Slope: When we think of “rise over run,” and look at the line, it does NOT rise at all. There isn’t an increase or decrease over the horizontal movement, at all. It is zero.
Formula: We can use the two highlighted points on the line in the formula. Here’s what we get:
If you remember from the Number Unit, zero divided by a non-zero number is zero. Division is asking a question about multiplication. 8 ÷ 4 asks, “four times what is eight?” Since the answer to that question is 2, 8 ÷ 4 = 2. Here we have 0 ÷ -5, which asks, “Negative five times what is zero?” Well, zero, of course!
Vertical Line
x = #
To the left we have the graph of the equation, which this time is NOT a function, x = 2. Every point on this line has an x – coordinate of 2.
Basic Meaning of Slope: When thinking of “rise over run,” and looking at this line, two things are confounding. First, there is no run, the input is only 2. (The domain is only 2.) Second, the line is unbound vertically. It goes up and down forever.
We cannot discuss rise over run here because there is no run.
Formula: If we plug the coordinates on the graph into the formula, this is what we get:
If you remember from the Number Unit, you cannot divide by zero. There is not an answer that is a number, of any kind, to this question. 6 ÷ 0 asks, “What times zero equals six?” You can check your multiplication tables if you like, but there isn’t an answer to the question. That’s why we call it undefined. Undefined means, literally, there is no meaning, no definition. In that sense, it is an ironic word.
Let’s summarize slope (so far).
- Slope is the rate of change
- It shows us how consecutive outputs change compared to consecutive inputs
- Slope is “RISE over RUN.”
- That is y over x, literally in a fraction.
- Lines are read from left to right.
- A positive slope increases from left to right.
- A negative slope decreases from left to right.
- A vertical line has an undefined slope.
- A horizontal line has a slope of zero.
- Three ways to find slope:
- If given a graph, count the slope.
- If given two points, use the formula.
- If given a formula, find either m or –B/A.
