Hello again, I am Mr. Tarrou! Whether you

are in algebra, pre algebra, geometry, algebra 2, or even precalculus, and calculus you need

a good understanding of slope and equations of lines. We are going to start out with some

equations of lines, finding slope. We will not be doing any application problems today

but the basic structures. Let’s first start with slope. We have a line over here with

a couple of points highlighted. We are going to use these two dots to find the slope or

the direction this line travels. Before you do that though you need to know a good definition

of slope. Slope is indicated with a cursive lower case m, is rise over run. This definition

is just three words. Everybody remembers this and it is really nice if you are just going

to look at a picture and tell what the slope is. You read graphs by the way left to right.

And as I read this left to right it is rising so I know when I am done that I am going to

have a positive slope. If the graph is falling down to the right then we have a negative

slope. If my line were horizontal my slope will be zero. And if my line is vertical that

will have an undefined slope. We are going to take a look at one of those later with

some numbers. Now we are going to look at this line. You have a couple of points. Start

with the left one first. How do you get from this point to this point? Visually you just

count. How far up do you have to go? This is the rise value. The rise value is one,

two,…You start here and don’t start counting when you touch the point. One, two, three,

four, five, six, seven, eight, nine, ten. We have a rise value of ten in this graph.

How far do you run over, how far to the right do we travel? The run value starting from

this point where I left off, one, two, three, four, five. So I have a rise value of ten

and a run value of five. So let me pick up this chalk that fell. Put this back in the

equation and we have the rise of ten over the run value of five. This gives us a slope

of two over one or two. Dividing by one does not do anything so we do not have to show

that in the final answer. Well that is fine but a lot of times in textbooks they don’t

give you a graph, they give you a couple of points. Let’s make sure you know how to find

the coordinates of a point. From the origin I am moving left three places and I am going

down what looks like four places. When you move to the left on the x axis, that is a

negative movement of x. That means this is negative three. If I go down on the y axis,

down is a negative movement, so that is negative four. We are going to label this as x sub

one and y sub one. And we are just going to find the slope again even though we already

know the answer is two. This point is two, going from the origin we are right two places.

That is a positive movement on the x axis. And we are going to go up one, two, three,

four, five, six. So we have a coordinate of (2,6) I already labelled this point as the

first point, so I am going to label this one the second point. If you are just starting

off in algebra you want to get into the habit of putting this x 1 y 1 and x sub 2 and y

sub 2 into your drawing. This will make sure you don’t put the numbers in the wrong spot.

Well rise over run doesn’t work very well if you have a bunch of numbers given to you,

especially without the picture. So another way of defining slope is y sub two minus y

sub 1 over x sub two minus x sub one. Your change in y is far up you went and the change

in x is how far over to the right you went. Let’s go back to these values and see what

we have. Y 2 is six minus, now my four is negative and there is a minus sign in the

equation so make sure you are using your parenthesis when you plug in your numbers, six minus negative

four. Six for my Y2 minus my other y value of negative four. Please note the minus that

is in the equation originally is right there, this negative is coming from my negative coordinate

of negative four. All of this is over X2-X1. My x sub two is two, I have got it well labelled,

minus again my minus sign is coming from the equation and not from the coordinate and always

use parenthesis when you substitute, minus negative three. If you have this written out

correctly we will have our slope. Six minus a negative… Well negative one times another

negative number and that is going make this cancel out. Six plus four is ten. Same thing

down here, we have two negatives touching. There is no division, there is no exponents,

we just have two negatives touching and that is multiplication. Negative times negative

again cancel out and two plus three is equal to five. We are once again getting down to

ten divided by five which was up here, again getting our answer of two. Now we are all

really happy with the fact that this line has a slope of two, we showed that twice.

What if you were told that you had a problem whose slope, this is going to be another example,

whose slope was equal to two and that goes through the point of….. I am going to come

back over here and I am going to pretend that I don’t know that this y value is negative

four. Again we are going to another example off of this same picture. So you have a slope

of two and point of negative three and this unknown value of y. We know it is really negative

four:) We have another point this line goes through of two six. So I gave you a slope

and I am giving you a question of, “What is the y value of this coordinate?” In all math

classes if they give you distance you are going to use the distance formula. If you

are given an area you will use that area formula. If they gave you slope and then asked you

a question, guess what, you are using the slope formula! So, I am going to walk through

this again. I am going to keep my labels of x sub 1 y sub 1 and x sub 2 and y sub 2. We

are going to walk through the slope formula just like we did a second ago. That is going

to be the slope is equal to two equals Y2 is 6 minus y over X1 of 2 minus don’t forget

the parenthesis and negative three. We are going to find that y value of negative four

in a minute. What do we do from here? Well, we have two equals six minus y over two, the

negatives are touching there is no exponent or weird stuff going on, just subtracting

a negative three so that is just two plus three is five. And we have the y here on the

right hand side of the equation, I want that alone. I want to find that is negative four

in a minute. With that y there is a division of five and I want to get rid of that division

of five. If you have a division of five and you want to get rid of it, if you are solving

equations you want things to go to the other side of the equation….I want that five to

go from the right to the left hand side, you must do the opposite of how it is connected

to the variable. The five is connected with division so I am going to get a different

color chalk here and we are going to multiply both sides of the equation by five. I am going

to get an answer of five times two is ten equals six minus y. I am almost running out

of room here. I am going to have erase a little bit down here so I can keep going. Now I have

got the y and a six. There is a positive six here, the minus sign is for the y, and this

six is positive. I want to get rid of the positive six with the x so I am going to do

that by subtracting both side by six, and giving my self a little more room to write

up here, ten minus six is equal to four and the six is now gone because I subtracted it

so negative y. I still have not figured out that y is negative four yet because y is not

yet by itself, it has a negative sign. So we are going to divide both sides by negative

one to get rid of that and for the third time we know the y value of that point that I just

erased, that you can back up the tape and look at, is negative four. So if a textbook

gives you a slope and a couple of points but you are missing one of those points, use the

slope formula and you will get your answer. Ok, now with slope, a lot of times you finding

the equations of lines. So let’s do one of those problems. Let’s say we want to find

the equation of a line that is passing through the point four seven and eight ten, and I

want the equation again. This is going to be pretty hard if you do not know don’t know

the three or four standard forms of a line. Right now I am going to bring up three and

uhh… let’s just say there is three. We have slope intercept form which is y equals mx

plus b. We have got Point Slope which I will be using in a minute which is y minus y sub

one, this is where we are going to put a number in a minute, equals m times x minus x sub

one. Again we are going to put some numbers in here in a minute for slope and these values

of x and y that we are given in the point. Another form or equation of lines is Standard

Form. This is Ax plus By plus c equals zero. A, B, C, these all have to be integers and

A should be positive. We are not going to work with this right now though. We are mainly

going to focus on this Point Slope Form. The name of the equation tells you what you need.

If you want the equation of a line you need a point and a slope. I have two points but

I do not have a slope. So let’s figure out what that is. X1 and Y1. X sub 2 and Y sub

2. Slope is Y sub two minus y sub one over x sub two minus x sub one. That is three.

That is four, and that is my slope. Now we are going to use that slope and one of these

points, and it makes absolutely no difference which one I pick. So I am going to pick the

first point just because it is the first thing I looked at. There is my point and there is

my slope and there is point slope form of a line. We are going to find the equation

of this line passing through this point using this slope. Y minus Y1 and the y of my point

is seven equals m, that is where this three fourths goes times x minus X1… my X1 equals

four. Now we have an equation with a fraction in it and a lot of people don’t like those

fractions. You can deal with it. This has a division of four and this has a value of

four so really it is not going to be a big problem at all. If however you are one of

those students that really really really do not like fractions you can get rid of it right

from the get go. This is probably not a good idea because this will cancel out, but lets

go ahead and multiply both sides by four. Now what is the purpose of this. A fraction

bar is division. If you have division that you don’t want, multiply it out. These parenthesis

are locking all of this together as one term so I only have to multiply by four once. When

I multiply by four these fours just cancel out. Over here the four is going to be distributed

through the parenthesis. We end up with four y minus, four times negative seven is negative

twenty-eight equals three times x minus four. I am going to run out of space again so I

am going to erase these standard forms here for you…for myself so I can write this out.

I am going to take the three and multiply it through the parenthesis. We get four y

minus twenty-eight equals three x minus twelve. I want to get y by itself. That is easy to

graph when you are in slope intercept form. Which I will have to do in another video.

This y I am going to isolate so I need to move this twenty-eight. It is minus twenty-eight

and whenever you want something to go over the equal sign you need to apply the inverse

operation, so I am going to add twenty-eight. You get four y equals three x plus sixteen.

And now I have a four that I do not want. So again if I want to graph this, if i want

to get this in slope intercept form, which is when y is by itself y=mx+b… so I am going

to divide everything by four. Y is equal to 3/4, there is the fraction that we got rid

of in the first place… it is coming back again, but for slope this is rise over run

and we want this fraction plus sixteen divided by four is four. This is our equation. We

started off with point slope form and what you just saw me do was convert this into slope

intercept form. This is very nice for graphing because we have the slope and the y intercept.

I think I am out of time. So, I will be back with you in a minute.