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Lesson 3: Slope

# Slope review

The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

## What is slope?

Slope is a measure of the steepness of a line.
$\text{Slope}=\frac{\text{rise}}{\text{run}}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$
Want an in-depth introduction to slope? Check out this video.

### Example: Slope from graph

We're given the graph of a line and asked to find its slope.
The line appears to go through the points $\left(0,5\right)$ and $\left(4,2\right)$.
$\text{Slope}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{2-5}{4-0}=\frac{-3}{4}$
In other words, for every three units we move vertically down the line, we move four units horizontally to the right.

### Example: Slope from two points

We're told that a certain linear equation has the following two solutions:
Solution:
Solution:
And we're asked to find the slope of the graph of that equation.
The first thing to realize is that each solution is a point on the line. So, all we need to do is find the slope of the line through the points $\left(11.4,11.5\right)$ and $\left(12.7,15.4\right)$.
$\begin{array}{rl}\text{Slope}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}& =\frac{15.4-11.5}{12.7-11.4}\\ \\ & =\frac{3.9}{1.3}\\ \\ & =\frac{39}{13}\\ \\ & =3\end{array}$
The slope of the line is $3$.

## Practice

Problem 1
What is the slope of the line below?
Give an exact number.

Want more practice? Check out this Slope from graphs exercise and this Slope from points exercise.

## Want to join the conversation?

• I dont understand this slope thing at all
can you help me
• You can always figure out the slope of a line if you have 2 points. If you are not given 2 points, you can find 2 points on the graph and use them to find the slope.

Here are some good things to know:
- m = slope
- (x₁, y₁) = point 1
- (x₂, y₂) = point 2
- rise = the difference in the y-values (y₂ - y₁)
- run = the difference in the x-values (x₂ - x₁)

In the slope formula, the slope (m) is equal to rise over run:

m = rise / run
= (y₂ - y₁) / (x₂ - x₁)

Let's say we are given a line with points (4, 2) and (6, 1). If we say that point 1 is (4, 2) and point 2 is (6, 1), then:

x₁ = 4 and x₂ = 6
y₁ = 2 and y₂ = 1

Now we just need to plug these values into the slope formula:

m = rise / run
= (y₂ - y₁) / (x₂ - x₁)
= (1 - 2) / (6 - 4)
= (-1) / 2
= -1/2

So the slope (m) is -1/2.

The main thing to keep track of is which point is (x₁, y₁) and which point is (x₂, y₂). You don't want to mix these up.

A few tips for graphs of slopes:
- a perfectly horizontal line has no slope
- a perfectly vertical line has a slope that is not defined
- a line that goes upwards (from left to right) has a positive slope
- a line that goes downwards (from left to right) has a negative slope

Hope this helps!
• Are you supposed to simplify 4.5 by 1.5??
• for the sake of the equation make 4.5 into 45 and make 1.5 into 15. From here, simplify 45/15. once you are done simplifying, 45/15 into 3/1, you have to reverse it to make it into the correct answer for slope equations (difference of y/difference of x).
• Why the formula for slope is △ y/ △ x not △ x/ △ y?
• Here are my thoughts...
Slope is a measurement of how steep the line is. The steepness is determined by how fast the line rises/falls. Thus, the predominate measure of importance is the change in Y. If you had △ x/ △ y, then the measurement is more descriptive of how fast the line is moving left/right, which doesn't really describe steepness.
• why would you have to do the close one said it passes throw 0,5 and 4,2 why can't you just do it were it passes throw the x and y
• You can calculate the slope from the x- and y-intercepts. But in this case, we can't tell exactly what the x-intercept is just from looking at the graph - we can only see that it is somewhere between 6 and 7.

In order to accurately calculate the slope, we need to use points where we know the exact value. We know the exact value of every point on the grid where the graph lines intersect. So when the line crosses one of those points, we know the exact coordinates for that point on the line.

For example:
(-4, 8)
(0, 5)
(4, 2)
(8, -1)

Any of these points may be used to calculate the slope, and you should get the same answer no matter which 2 points you use.

Hope this helps!
• How can it be a fraction, because when I put in a number that we used in class, it said that what I put in was wrong. If you guys could explain that concept to me that would be greatly appreciated
• The slope of a line is defined as a fraction: rise over run; or (y2-y1)/(x2-x1). So slope is always a fraction. Even if you get a number like 5 as a slope, you need to change it into 5/1 (fraction form) to understand what it means related to the lines movement.
• This is genuinely so much easier to learn than the way my tutor taught it.
• thats kinda mean
• how do you graph a linear equation in slope intercept form such as:
y= 1/2 x - 3
• Slope-intercept form: y=mx+b where...
m = the slope
b = the y-intercept at the point (0, b)

Start by graphing that point.
Then use the slope to find more points.
Your slope = 1/2. This means from the point (0, -3), go up 1 unit and right 2 units to find the next point. You can repeat this as many times as you want to find points on the line.
Don't forget to actually draw the line.

FYI... Search for the leason on graphing from slope intercept form. There is a search bar at the top of all KA screens.
• I was wondering if you only have lets say, 2 points on a line, and I was wondering if you could solve for the slope, even if thats all you have
• Two points would be enough.
Your points would be (x1, y1) and (x2, y2) then the slope m would be:
m = y1 - y2 / x1 - x2