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# Intro to slope

Walk through a graphical explanation of how to find the slope from two points and what it means.
We can draw a line through any two points on the coordinate plane.
Let's take the points $\left(3,2\right)$ and $\left(5,8\right)$ as an example:
The slope of a line describes how steep a line is. Slope is the change in $y$ values divided by the change in $x$ values.
Let's find the slope of the line that goes through the points $\left(3,2\right)$ and $\left(5,8\right)$:
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{6}{2}=3$
Use the graph below to find the slope of the line that goes through the points $\left(1,2\right)$ and $\left(6,6\right)$.
$\text{Slope}=$

Notice that both of the lines we've looked at so far have been increasing and have had positive slopes as a result. Now let's find the slope of a decreasing line.

## Negative slope

Let's find the slope of the line that goes through the points $\left(2,7\right)$ and $\left(5,1\right)$.
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{-6}{3}=-2$
Wait a minute! Did you catch that? The change in $y$ values is negative because we went from $7$ down to $1$. This led to a negative slope, which makes sense because the line is decreasing.
Use the graph below to find the slope of the line that goes through the points $\left(1,9\right)$ and $\left(4,0\right)$.
$\text{Slope}=$

## Slope as "rise over run"

A lot of people remember slope as "rise over run" because slope is the "rise" (change in $y$) divided by the "run" (change in $x$).
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{\text{Rise}}{\text{Run}}$

## Let's practice!

Heads up! All of the examples we've seen so far have been points in the first quadrant, but that won't always be the case in the practice problems.
1) Use the graph below to find the slope of the line that goes through the points $\left(7,4\right)$ and $\left(3,2\right)$.
$\text{Slope}=$

2) Use the graph below to find the slope of the line that goes through the points $\left(-6,9\right)$ and $\left(2,1\right)$.
$\text{Slope}=$

3) Use the graph below to find the slope of the line that goes through the points $\left(-8,-3\right)$ and $\left(4,-6\right)$.
$\text{Slope}=$

4) Use the graph below to find the slope of the line that goes through the points $\left(4,5\right)$ and $\left(9,5\right)$.
$\text{Slope}=$

5) Use the graph below to choose the slope of the line that goes through the points $\left(3,2\right)$ and $\left(3,8\right)$.
$\text{Slope}=$

## Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with a slope of $5$ is steeper than a line with a slope of $\frac{1}{2}$

7) A line with a slope of $-5$ is steeper than a line with a slope of $-\frac{1}{2}$

## Want to join the conversation?

• How can the slope value (1/2 or 5) be used in real life, and how can we use it in math?

Thanks!
• It could be used to simulate the steepness of a mountain/hill.
• why does math have to be so confusing?
• If you work hard then eventually math won't be as confusing!
• My dad left to get milk. He never came back :)
• My mom gets milk every week, and she always comes back! I confuse.
• Genuine question- when will i EVER use this IRL?
• no but you will for your math class so
• Can somebody tell me how to easily visualized. which slope is steeper?
• Try to think about it like this, imagine you are running up (positive slope) or down (negative slope) a flight of stairs.

If the slope is a larger number, than it is the same as taking several steps at once.
If the slope is a smaller number, it is as if you are taking less steps at once, not going up the flight of stairs as quickly.

Imagine a slope of 1, means for ever step you take with your feet you only go up one stair.

Imagine a slope of 5, this means for every step you take you go up 5 stairs. You get to the top and rise much quicker.

___________

Another visual example: Imagine you are going skiing. As you go down a slope, you expect the slope to be negative. You come from up high on the y axis and go down.

If now you go to a ski slope at -5 that means for every meter you glide forward on your skis towards to bottom of the hill (x-axis), you also go down 5 meters on the hills height (y-axis). This is very steep as you can imagine.

Now imagine you are going down a ski slope of only 1/2. This means for every 1 meter you glide forward on your ski you only get 1/2 meter further down the hills height.
• lmao what is y=mx+b gonna do for me in life
• ma’am I do not get this🧍🏻‍♀️
• I ain’t understand
• Think of slope of a given graphed line in this way, and this is also very helpful when you want to calculate slope from a table or a formula.

Slope = y2-y1/x2-x1 for any two points, 1 and 2, on the line or in the table, or calculated in the formula.

In the opening graph of this page, the two points with the dots (though you could use any two points on the line) are (in standard x,y sequence):

Point 1:(3,2)
Point 2:(5,8)

From those points, we get:

y2-y1 = 8-2 = 6.
x2-x1 = 5-3 = 2.

Slope is now determined by y2-y1/x2-x1, or 6/2, or 3.

I could also do y1-y2/x1-x2 instead of y2-y1/x2-x1, reversing the order of the points in the calculation and, sometimes, that's helpful to keep the algebra simple and positive. But, once calculated through, both ways yield the same results.

For instance, in the case I am talking about, if we reverse the order of the points in our math, we get:

y1-y2/x1-x2 = 2-8/3-5 or -6/-2.

A negative divided by a negative yields a positive so -6/-2 = positive 3.

That's the same result so I've proven the order of the two points in our calculation doesn't matter; it only requires that we keep the x and y order consistent throughout the calculations.
• So it doesn't matter then, what numbers you use for slope. What about when they don't give you a graph and make you find slope? I'm confused
• You haven't said what information you were given instead of a graph.

If you are given two points on the line, you can calculate the slope using the slope formula.

If you are given the equation of the line, you can:
-- Change the equation into slope-intercept form: y=mx+b and "m" will be the slope.
-- Or, you can calculate two points using the equation and then use the slope formula to calculate the slope.

All these options are covered in later videos.