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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 parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 5, comma, 8, right parenthesis 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 parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 5, comma, 8, right parenthesis:
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, 6, end color #e07d10, divided by, start color #1fab54, 2, end color #1fab54, end fraction, equals, 3
Use the graph below to find the slope of the line that goes through the points left parenthesis, 1, comma, 2, right parenthesis and left parenthesis, 6, comma, 6, right parenthesis.
start text, S, l, o, p, e, end text, equals

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 parenthesis, 2, comma, 7, right parenthesis and left parenthesis, 5, comma, 1, right parenthesis.
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, minus, 6, end color #e07d10, divided by, start color #1fab54, 3, end color #1fab54, end fraction, equals, minus, 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 parenthesis, 1, comma, 9, right parenthesis and left parenthesis, 4, comma, 0, right parenthesis.
start text, S, l, o, p, e, end text, equals

## 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).
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, start text, R, i, s, e, end text, end color #e07d10, divided by, start color #1fab54, start text, R, u, n, end text, end color #1fab54, end fraction

## 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 parenthesis, 7, comma, 4, right parenthesis and left parenthesis, 3, comma, 2, right parenthesis.
start text, S, l, o, p, e, end text, equals

2) Use the graph below to find the slope of the line that goes through the points left parenthesis, minus, 6, comma, 9, right parenthesis and left parenthesis, 2, comma, 1, right parenthesis.
start text, S, l, o, p, e, end text, equals

3) Use the graph below to find the slope of the line that goes through the points left parenthesis, minus, 8, comma, minus, 3, right parenthesis and left parenthesis, 4, comma, minus, 6, right parenthesis.
start text, S, l, o, p, e, end text, equals

4) Use the graph below to find the slope of the line that goes through the points left parenthesis, 4, comma, 5, right parenthesis and left parenthesis, 9, comma, 5, right parenthesis.
start text, S, l, o, p, e, end text, equals

5) Use the graph below to choose the slope of the line that goes through the points left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 3, comma, 8, right parenthesis.
start text, S, l, o, p, e, end text, equals
Choose 1 answer:
Choose 1 answer:

## 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 start fraction, 1, divided by, 2, end fraction
Choose 1 answer:
Choose 1 answer:

7) A line with a slope of minus, 5 is steeper than a line with a slope of minus, start fraction, 1, divided by, 2, end fraction
Choose 1 answer:
Choose 1 answer:

## 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!
(32 votes)
• It could be used to simulate the steepness of a mountain/hill.
(9 votes)
• why does math have to be so confusing?
(19 votes)
• My dad left to get milk. He never came back :)
(15 votes)
• that sad where did he go
(1 vote)
• lmao what is y=mx+b gonna do for me in life
(7 votes)
• NOTHING in reality unsless u have a job that includes math.
(3 votes)
• Is there another way of finding a slope without a graph?
(2 votes)
• It should give you points on the graph, use the formula y2-y1/x2-x1, to find the slope, (y2 a y-coordinate, of one of the point, you are not multiplying anything)
Hope this helps :)
(11 votes)
• What does steeper mean, greater slope? In that case, shouldn't a slope of -1/2 be steeper than -5, as -1/2 is greater than -5?
(3 votes)
• We are seeing absolute value, which means if it's negative, we have to see the more negative one as the greater one. If it's positive, we have to see the more positive one as the greater one(like we always did)
The symbol for absolute values are like so: | insert number |

Try these problems as examples:
Put > < = in the blanks.
a) | -6 | _ | -1 |
b) | 8 | _
| 9 |
c) | -4 | __ | -12 |

The answers are
a) > b) < c) <
(7 votes)
• Can somebody tell me how to easily visualized. which slope is steeper?
(7 votes)
• you know that 5 is greater than 1/2
so -5 is greater than -1/2
(1 vote)
• is there an easier way to find the slope?
(3 votes)
• yes, actually if you align the coordinates
For example: (3,2)
(5,8)
And then you would find the difference between the the numbers such as the difference between the 3 and 5 is 2, while the difference in the 2 and 8 is 6. After you get these numbers you can put them in the fraction form as you would after you find the numbers on the graph.This would then equal 6/2 or in simplified form = 3.
I hope this was helpful (:
(2 votes)
• Will the formula rise ovr run always stay the same or will it change? Helppp!
(2 votes)
• Not sure what you mean by it changing. Rise/run is a informal way of saying change in y/change in x. So since this is the definition of slope, yes it will always stay the same for linear equations.
(2 votes)