Hey, guys.
Welcome to Algebra 1.
Today's lesson is going to focus
on calculating slope
when you're given
two points on a line.
Your knowledge
of the coordinate plane
and what you learned in
pre-algebra
will help you
get through this lesson.
You ready to get started?
Let's go.
Okay.
Slope, you probably learned in
algebra or pre-algebra,
is a rate of change;
it describes a rate.
Often we'll have
points in a table.
You can interpret the slope
or a rate that way.
That's how we'll start out,
to get your mind warmed up
for dealing with slope.
"Melissa is training
for a marathon.
The table below shows
a record of her best times."
We see, in 15 minutes,
she ran 3 miles;
30, she ran 6 miles;
45, she ran 9 miles, okay?
So slope
or the rate of change--
let's say we wanted to see
if it was positive
or negative.
Is Melissa getting better
or not as she's training?
Let's see if her
distance increases
as her time increases.
At 15 minutes,
she ran 3 miles.
When the minutes
increase to 30,
her miles increase to 6.
When the minutes increase to 45,
her miles increase to 9.
Okay, all right.
It looks as though
she is increasing.
Her rate of change,
or in her training, her slope,
it's getting better
every time.
Let's look at this graphically
and see what this looks like.

(female narrator)
On the graph,
the X-axis has minutes,
and the Y-axis has miles.
If we plotted those points
on our coordinate plane--
I got the first quadrant here
since things
were positive--
we see that it looks like--
if I do just
a rough line--
it is increasing.
If you look at it
graphically,
or if Melissa looked at this
graphically,
she'll see she's doing good
and should keep training.
As her minutes increase,
the miles she runs increases.
We'd say the slope here
is positive.
That's a way we can
describe slope,
or rate of change.
There are four descriptors;
let me show you those.
The slope of a line
can be called positive.
Like we read words
from left to right--
like when I read "slope,"
I start at the "S"--
we read graphs
left to right.
From left to right,
this line rises.
That's a positive slope.
From left to right,
this line falls.
It has a negative slope.
From left to right,
there's no change in this line,
so we say it's a zero slope.
This line doesn't actually
go left to right;
it goes up and down.
This slope is undefined.
I'm going to flip back
a second.
We have a positive slope,
a negative slope,
zero slope, undefined slope.
Those are the four descriptors
for slope.
Many times, you want
to actually get a value.
So I'll show you
how to get that rise over run,
which is also
what we describe slope as.
Slope is the change in Y...
divided by the change
in X.
Another way we describe that,
because the Y axis
goes up and down
and the X axis
goes left to right,
is we also say that that
is the "rise over the run."
When you have to determine
the slope of a line from a graph
and you want to count out
the slope, you'll do this.
Start at the point
furthest to the left.
In this case,
this is the point is at (1, 1).
I'll count how far
I have to rise up,
then run over
to get to that second point.
Write it like a ratio,
or a fraction,
and that's my slope.
I'll start from this point.
I have to rise two
to get to the same level
as that second point...
and I have to run four
to get over
to that second point.
So my rise was 2,
my run was 4.
I'd say, "All right,
my slope is 2 over 4."
Or reduce, because we always
like things in simplest form--
slope is one-half,
and that's it.
That's how you determine
the slope from two points
that are already on a graph,
already plotted for you.
Let's try another one.
Remember we read from left
to right on the graph.
The left point, the point
that's furthest to the left,
is up in the second quadrant,
and this second point's
down here in the third.
Here, I don't have a rise,
I have a fall.
You can visualize
that that line
passing through those points
has a negative slope.
I'll start with that point
furthest to the left.
You don't have to
start with that point,
but if you do always start
with the point
furthest to the left,
it's just more likely
you won't make
a mistake while you count.
It'll be easier figure out
if a slope
is positive or negative
if you get in the habit
of reading from the left.
So we're going to read
this graph from the left also.
Here, we're going to start
with the point furthest left.
Here it's a fall;
I've got to drop down.
So one, two, three,
four, five...
and I went over one.
So if my slope
is the rise over run,
in this case I had a fall,
so I'll say the slope
is negative 5 over 1.
I could represent that
as a whole number
because the denominator is 1,
and say that my slope
is negative 5.
Either answer is acceptable.
Either one
is perfectly fine.
Okay, I want you
to try this one.
Press pause
and take a few minutes,
and count out the slope
for that rise over run.
To check your answer,
press play.

(female announcer)
What is the slope?
The graph
has two plot points.

The first is at
(negative 3, 1).

The other is at
(negative 2, negative 1).
Okay,
let's see how you did.
If you started
from that left point,
that far left point,
we're going
to go down one, two.
Over one, two,
three, four, five.
If slope is rise over run,
in this case is was a fall,
so negative 2 over 5.
That would be your slope
for this problem, all right?
Let's keep going.
There's another way
that you can determine
the slope given two points;
let's use this example.
"What is the slope of the line
"that contains
these given points?

(12, negative 2) and (9, 3)?"
We could get
some graph paper
and we could plot
these points, okay?
Another way to determine
the slope of a line
is to use the slope formula.
Let me show you that.
Given two points--and I'll
explain these symbols soon--
but the slope, M--
"M" stands for slope--
the slope, M, of a line given
points (X sub-1, Y sub-1),
and (X sub-2, Y sub-2)--
so that's just saying
given two different points--
the slope is M equals
Y2 minus Y1
divided by X2 minus X1.
That's another way
that you could calculate
the slope of a line.
That's your rise over your run
written as a formula.
Let me show you
how to work this
by jumping back
to example three.
They gave us two points.
First thing I'll to do is write
the slope formula up here.
So M equals Y2 minus Y1
over X2 minus X1.
So I'm going to label
the first point X1, Y1.
I'm going to label
the second point X2, Y2.
Just labeling it
will help you
keep everything straight,
okay?
Now that I have
the points labeled,
I'm going to use
the slope formula.
So M equals Y2 minus Y1,
so that's 3
minus negative 2...
over X2 minus X1.
So that's 9 minus 12.
And then we're just going
to simplify this.
You're seeing
your numerator,
you have that double negative.
From pre-algebra you remember
that just becomes a plus.
So 3 plus 2, that's 5.
And 9 minus 12,
that's negative 3.
The slope of the line that
contains these two points,
negative five-thirds.
It's the same answer
if you plotted these points
and counted the rise over run.
You should be familiar
with the slope formula
because you may be asked,
specifically,
to use that method
to find the slope.
All right,
let's try another one.
We have (8, 5) and (12, 5).
Calculate the slope
of the line
that contains
the given points.
I'll write formula up here.
So M equals Y2 minus Y1
over X2 minus X1.
I'm going to label
my points.
So (X1, Y1), (X2, Y2).
Now let's start
substituting.
So I need Y2 minus Y1,
so that is 5 minus 5...
over X2 minus X1,
so that's 12 minus 8.
So 5 minus 5, that's 0.
12 minus 8, that's 4.
0 divided by 4 is 0.
So the slope
of this line is 0.
If you remember when we looked
at the slopes graphically,
that's that flat line,
that horizontal line.
You could get a visual
that the line
that contains these two points
is a horizontal line.
All right,
let's keep going.
And it is your turn.
Press pause,
take a few minutes
to work through
these problems.
I want you to be comfortable
using a slope formula.
When you're ready to check
your answers, press play.

(female narrator)
Use the slope formula to find
the slope of the line
that contains the given points.

Number 1: The points are
(negative 1, 2) and (1, 10).
Number 2: The points
are (7, 3) and (7, 9).
All right,
let's see how you did.
For the first one,
the slope of that line
that contains
those two points...
was 4, or maybe
you wrote 4 over 1.
For number two,
the slope of the line
that contains those two points
is undefined.
Okay? All right.
Let's look at these to show you
the work for these.
Let me write
the slope formula.
Let's get the pen back.
So M equals Y2 minus Y1
over X2 minus X1.
And I'm going
to label my points.
So X1, Y1; X2, Y2.
Now I'm just going to
substitute into the formula.
So Y2 minus Y1,
that'd be 10 minus 2...
over X2 minus X1,
so that's 1 minus negative 1.
So 10 minus 2, that's 8.
1 minus negative 1,
that's going to become a plus.
So in other words,
1 plus 1, that's 2.
Then, 8 divided by 2 is 4.
So the slope
of that line is 4.
Or visually,
that would be a line...
that went up, that would rise
from left to right.
That second one,
that one that was undefined.
Let's write our slope formula.
So Y2 minus Y1
over X2 minus X1.
Let's label our points,
so X1, Y1; X2, Y2.
Now, let's substitute
into the slope formula.
So Y2 minus Y1,
that'd be 9 minus 3
over X2 minus X1,
so that'd be 7 minus 7.
So 9 minus 3, that's 6.
7 minus 7 is 0.
It's not possible
to divide any number by 0.
If you put 6 divided by 0
in the calculator,
you'd probably get an error
saying "undefined"
or "no domain."
That's how I know
that this slope is undefined.
Don't leave
the answer as 6 over 0.
Write the answer
as "undefined."
Remember, the slope
of a line that's undefined,
or a line
with an undefined slope,
it's a vertical line,
straight up and down.
Okay?
All right.
Great job solving problems
involving calculating
the slope of a line
by counting the rise over run
on the graph
or using the slope formula.
Hope to see you soon
for more Algebra 1. Bye!