Since pictures are worth a million words, and they're going to be especially
useful in predicting outcomes...
I'm going to ask you to take an equation a linear equation and come up with the
picture of it at which would be a line
do you remember when I told you that if we took all the points that fit into an
equation or that were the solution to an equation here are four of them and
plotted them
do you remember what happens let's plot 45
over four and up five one negative one over one and down one and 1.50
i'm going to go over one and a half squares and up none
see that over one and a half squares and up no squares and - 10 / - and up one
now they're spread out differently but one thing's for sure if i did other
points
Cody they all line up every pair that fits into this equation that is a
solution to this equation is on that line and in fact that line goes on how
far no one better day and beyond goes on forever
so you can predict the future almost of this equation which can represent some
very useful things in life with a drawing of the line now if you're trying
to find that line all you really need is two points
here's these points i want to demonstrate something to you
doesn't matter which two points i pick if I picked these two and i didn't even
pay attention to the other two
I don't know where the line is wooden I what if I picked these two same line
what if i pick these two it doesn't matter i really only need two points
I'm going to tell you to do three though ok so here's a kind of problem i'm going
to give you a graph this line draw the line that represents this equation
Wow
I know there's an infinite amount of points on the line here and i know you
only need to
and as i said i'm going to ask you to find three of them and i'm going to ask
you to make a chart
now you get to pick any numbers to put in and I suggest that you put them in
for X so let's make them easy numbers
the easiest number i know is 0 so i always use 0 put in 0 for X and nicely
I can find why let's see y equals 2 times 0 which is 0-1 so 0-1 in this case
y is negative 1 so the pair 0 negative 1 is a solution and I'll graph it i'll go
over none and down one
now i need at least one more what's the next easiest number you can think of
let's put in one put in 1 for X 2 times 1 is 2 minus 1 ends up in this case why
comes out one so i have the pair 11
so start at the origin go over 1 and up one
and you've got the point you probably already know where the line is
but i always tell people to do one more point in the next easiest one I know
doesn't matter but we're going to use the next easiest number is to two times
two is four and four minus 1 is 3 y is 3
so let's start at the origin go over to and up three now before we do it let's
pray that they line up and i can see that they do
the reason i have you do three points and not to go as we do three and they
don't line up you can see what you let you know when you made a mistake
so always do three points and if they line up like this
you know you have the right answers now you have that we still need the line and
what do we do one better day
they all connect them and add the arrows and we know where the line is the mobile
let's do another one I hope you noticed in the last one that was way easier to
find why if we began by getting Y by itself and I'm going to tell you to
always do that
oh I'm going to make your chart and begin by solving for y
first you just see that mine for X is keeping Y from being alone so i'll
subtract 4x from both sides talk and I'll get eight minus 4x
now who's keeping wife and being alone - so I'll / - everybody and I'll get y
equals
4 - 2 X now it's going to be much easier that way and there's going to be way
more reasons in the future for solving for y so get used to for cripes sake
solve for y now we're back to the same old way
put in various numbers for X let's put in the three easiest I know 0 4 minus 2
times 0 is 4-0 or for that I have a pair 0 for 0 / 0 and up for what's the next
number i should put in let's make life easy
I'll put in a 1 for x + 4 minus 2 times 1 is 4 minus 2 or two
so in this case or 14 x + 2 for y so I go over 1 and up to
I asked you to do three points remember so you can check your errors if you make
one so we'll put in what world
it wanted to do it for Christ's sake put an easy numbers not go ahead
we could put in negative 87 but we can also slap ourselves in the face
let's put in an easy number like to
let's see 4 - 2 x 2 or 4 - 4 we're going to get 0 0 can be an answer
I'm going to graph the pair 20 by going over to and up none i'm very happy about
this aren't I
because no one better day lineup beyond
I draw the line and put on the arrows and I have the graph if you would of
this linear line your equation
I got a tough one for you see if you can do it
we're going to graph this equation remember how to do it how do we begin
always begin to be glad in the future if you did that
now let's look at who's keeping Y from being alone
there's a minus 2x + a3 let's begin by adding 2x to both sides now i'm going to
divide everybody by three
be careful going to do it to everybody on the left side we just get why on the
right side we have 3 over 3 which is 1 and 2 over 3 which is 2 over 3
well it doesn't come out nice he had a fraction but I have a fix for that
that's why I wanted to do this example
usually we've been putting in the three easiest numbers that we know 0 1 & 2
well I going to do well I want to make life easy
you can put anything you want Oh No
instead of putting 0 1 into let's put in mobile with the oil billions is going to
make life easier
put in multiples of the denominator which is in this case three
so we're going to put in multiples of three and it's going to come out much
nicer
Oh watch gonna put in 0 of course that will cancel 2 times 0 is 0 and I get a
one that's easy / 0 and up one
now let's put in three that's a multiple of three
look the threes cancel that's why i put in a three and i'm left with 1 plus 2
oh that's a nice easy number and I'll graph that go over 3 and up three
and now let's put in i could put in six also but I chose to put in negative 3
and the threes are going to cancel with the negative isn't and I'm going to have
a negative 2 plus 1 which is a negative one graph that point
basically we're going to go back three and down one ball back three and down
one and son of a gun they line up
yeah and all you have to do is do all the line
no one better day i'll be on with the arrows on it
ok so make sure if you have a fraction use multiples of the denominator will
make life way easier for you
this one's a little bit different but it's still follow sometimes things are
so easy there hard still follows our rules
what's your first step
get used to that
who's keeping Y from being alone the three
he's multiplying so i'll divide
so in this case so y equals two
now as I said sometimes things are so easy there hard
there's no fraction here so I don't need multiples of anybody i'm going to put in
the three easiest numbers i know for X so what is the three easiest numbers
zero one and two not so bad
the problem here is there isn't a next to put in for but remember what we're
trying to get here we're trying to get the y coordinate so in the first . what
is why what is widened will told you right there
why is too I told you it's so easy it's hard
why is to didn't even have to put it in the graph that point
oh we get over none and up to in the second . guess what why is told you why
is to go over 1 and up to the third . what is why why is got it y is 2 and
nicely if we graph to to go over to and up to the mall
you're gonna feel a lot better you are correct sir they line up
this is called a constant function it's constantly -
but son of a gun it is a linear line your function
no one better day they are isn't it forms a line every time
why is to now the solve for y method is going to work
the vast majority of the times there is one situation when it won't work and I'd
like you to see that
houston we have a problem in this case when I asked you to begin with
all but there isn't a why so
Oh what do we do well you don't have to be a brain surgeon to know
oh whoa we don't have any other choice Bob right so we'll solve for x but
normally we're going to try to solve for y
there isn't a why so we'll solve for x who's keeping the X from being alone
the two so i'll divide by 2 i'll get x is minus 1
now as I said this one's a little bit weird but not difficult
what do we put in for X be careful here
what are we going to put in for X in R 3 points
I'll read the equation x has got to be negative 1 they said it was negative
didn't say anything about why since they didn't say anything about why why can be
anything and I get to pick why I pic
easiest number zero my graph negative 10 back one and up none of those there and
you have to pick the first points why
what are you going to pick for the second points why pick anything you want
the next easiest number one little back one and up one
starting to line up here
the third point no fraction so let's do to go back one and up to
I'm feeling pretty good they're lining up
no one better day and be a hard drive it's okay kid you okay we'll go practice
do your homework