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professor Dave again let's talk about

slopes and intercepts in math when we

use the word slope it doesn't mean we

are going to hit the slopes although

there is some similarity between the

downward slope of a mountain and the

meaning of this word in math it means a

lines rate of change in the vertical

direction which can be shallow or steep

just like a mountain to understand how

the slope of a line relates to the

equation of the line let's introduce a

common form for linear equations y

equals MX plus B we already know about x

and y these are the independent variable

and dependent variable respectively but

these other two terms represent

characteristics of the line M represents

the slope of the line which is as we

said essentially how steep the line is

and B represents the y-intercept which

is the y coordinate for the point where

the line crosses the y axis let's start

by looking at the simplest example

possible y equals x we already learned

how to plot this line so here it is

with every point showing an ordered pair

where the x and y values are equivalent

to calculate the slope we must

understand that this is equal to rise

over run which means change in Y over

change in X we can pick any two points

on this line and with these points we

take the difference in their Y values

and divide by the difference in their X

values Y is up and down or the rise and

X is side to side or the run rise over

run let's pick these two points two two

and four for rise over run can be

represented this way y2 minus y1 over x2

minus x1 so we just plugged the

coordinates in y2 is for y1 is 2 X 2 is

4 and X 1 is 2 we sin

bye and get 2 over 2 or 1

the slope of this line is 1 that should

make sense because y equals x is in y

equals MX plus B form it's just that M

is 1 and B is 0 1 X is just X and B is 0

because the line crosses the y axis when

y equals 0 so we get this very simple

equation y equals x if a line has a

slope that is greater than 1 it will

tilt upwards like this steeper and

steeper climbing towards infinity until

it reaches complete verticality at which

point the slope is undefined because the

line rises all the way to infinity in

the Y direction without any run at all

and infinity over 0 is undefined if

instead the slope is less than 1 the

line tilts downwards this way the slope

getting smaller and smaller until it is

completely horizontal at which point the

slope will become 0 the line runs all

the way to infinity in the X direction

without rising at all and 0 over

infinity is 0 so vertical lines have an

undefined slope horizontal lines have

zero slope and everything in between has

some slope between 0 and infinity if we

continue past to the horizontal the

slope becomes increasingly negative

because as we run in the positive

direction the rise is negative or moving

downwards the slope will increase in the

negative direction approaching negative

infinity until vertical where the slope

becomes undefined let's just quickly

mention that any two points define a

line you could draw any two points at

random on the coordinate plane and there

is a line that connects them however all

lines contain an infinite number of

points since the coordinates can become

infinitely precise and any point on the

represents an x and y value that

qualifies as a solution to the equation

we could pick two points like three two

and five seven and draw a line between

them we could also calculate the slope 7

minus 2 is 5 and 5 minus 3 is 2 so the

slope is 5 halves always remember that

when calculating slope it doesn't matter

which point is point 1 and which is

point 2 but the coordinates of each

point must remain together so whichever

y value is y to the x value in that

point must be x2 and not x1 that would

lead to an incorrect slope calculation

also one point and a slope can define a

line we could say that a line passes

through 2 4 and has a slope of 3 halves

to draw this line we would just go to

the point 2 4 and start applying the

slope up 3 over 2 this could also be

down 3 2 to the left because negative 3

over negative 2 also equals 3 halves and

that's how the slope which we can regard

as the rate of change makes sense

regardless of which direction you travel

now that we understand slopes and

y-intercepts let's check comprehension

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professor Dave explains at gmail.com

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