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Graphs of linear equations | Linear equations and functions | 8th grade | Khan Academy - YouTube
Channel: Khan Academy
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Let's do a couple of problems
graphing linear equations.
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They are a bunch of ways to
graph linear equations.
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What we'll do in this video
is the most basic way.
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Where we will just plot a
bunch of values and then
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connect the dots.
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I think you'll see
what I'm saying.
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So here I have an equation,
a linear equation.
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I'll rewrite it just in case
that was too small. y is equal
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to 2x plus 7.
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I want to graph this
linear equation.
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Before I even take out the graph
paper, what I could do
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is set up a table.
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Where I pick a bunch of x values
and then I can figure
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out what y value would
correspond to
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each of those x values.
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So for example, if x is equal
to-- let me start really low--
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if x is equal to minus 2--
or negative 2, I should
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say-- what is y?
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Well, you substitute
negative 2 up here.
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It would be 2 times
negative 2 plus 7.
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This is negative 4 plus 7.
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This is equal to 3.
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If x is equal to-- I'm just
picking x values at random
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that might be indicative of--
I'll probably do three or four
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points here.
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So what happens when
x is equal to 0?
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Then y is going to be equal
to 2 times 0 plus 7.
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Is going to be equal to 7.
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I just happen to be
going up by 2.
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You could be going up
by 1 or you could be
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picking numbers at random.
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When x is equal to
2, what is y?
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It'll be 2 times 2 plus 7.
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So 4 plus 7 is equal to 11.
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I could keep plotting
points if I like.
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We should already have
enough to graph it.
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Actually to plot any line, you
actually only need two points.
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So we already have one
more than necessary.
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Actually, let me just do one
more just to show you that
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this really is a line.
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So what happens when
x is equal to 4?
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Actually, just to not go up by
2, let's do x is equal to 8.
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Just to pick a random number.
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Then y is going to be 2 times 8
plus 7, which is-- well this
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might go off of our graph
paper-- but 2 times 8 is 16
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plus 7 is equal to 23.
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Now let's graph it.
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Let me do my y-axis
right there.
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That is my y-axis.
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Let me do my x-axis.
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I have a lot of positive values
here, so a lot of space
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on the positive y-side.
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That is my x-axis.
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And then I use the points x
is equal to negative 2.
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That's negative 1.
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That's 0, 1, 2, 3,
4, 5, 6, 7, 8.
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Those are our x values.
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Then we can go up
into the y-axis.
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I'll do it at a slightly
different scale because these
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numbers get large
very quickly.
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So maybe I'll do it in
increments of 2.
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So this could be 2, 4,
6, 8, 10, 12, 14, 16.
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I could just keep going
up there, but
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let's plot these points.
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So the first coordinate I have
is x is equal to negative 2, y
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is equal to 3.
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So I can write my coordinate.
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It's going to be the point
negative 2, 3.
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x is negative 2.
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y is 3.
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3 would land right over there.
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So that's our first one,
negative 2, 3.
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Then our next point.
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0, 7.
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We do it in that color.
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0, 7.
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x is 0.
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Y is 7.
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Right there.
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0, 7.
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We have this one
in green here.
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Point 2, 11.
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2, 11 would be right
about there.
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And then this last point-- this
is actually going to fall
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off of my graph.
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8, 23.
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That's going to be way
up here someplace.
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If you can even see
what I'm doing.
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This is 8, 23.
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If we connect the dots, you'll
see a line forms. Let me
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connect these dots.
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I've obviously hand drawn it, so
it might not be a perfectly
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straight line.
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If you had a computer do it, it
would be a straight line.
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So you could keep picking x
values and figuring out the
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corresponding y values.
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In the situation y is a function
of our x values.
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If you kept plotting every
point, you'll get every line.
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If you picked every possible x
and plotted every one, you get
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every point on the line.
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Let's do another problem.
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At the airport, you can
change your money from
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dollars into Euros.
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The service costs $5.
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and for every additional dollar,
you get EUR 0.7.
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Make a table for this and plot
the function on a graph.
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Use your graph to determine how
many Euros you would get
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if you give the office $50.
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I will write Euros is equal to--
so let's see, it's going
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to be dollars.
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So you're going to have
to give your dollars.
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Right off of the bat, they're
going to take $5.
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So dollars minus 5.
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So immediately this
service costs $5.
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And then everything that's
leftover-- this is your
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leftover-- you get EUR 0.7 for
every leftover dollars.
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You get 0.7 for whatever's
leftover.
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So this is the relationship.
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Now we can plot points-- we
could actually answer their
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question right off the bat.
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If you give them $50, we don't
even have to look at a graph.
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But we will look at a graph
right after this.
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So if you did Euros is equal
to-- if you have given them
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$50-- it would be 0.7
times 50 minus 5.
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You gave them 50.
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They took 5 as a service fee.
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So this is just $45 It would
be 0.7 times 45.
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I could do that right here.
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45 times 0.7.
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7 times 5 is 35.
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4 times 7 is 28 plus 3 is 31.
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And then we have only one number
behind the decimal,
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only this 7.
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So it's 31.5.
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So if you give them $50, you're
going to get EUR 31.5.
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Euros, not dollars.
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So we answered their question,
but let's actually do it
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graphically.
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Let's do a table.
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Maybe I'll get a
calculator out.
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I'll refer to that
in a little bit.
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So let's say dollars
you give them.
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And how many Euros do you get?
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I'll just put a bunch
of random numbers.
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If you give them $5, they're
just going to take
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your $5 for the fee.
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You're going to get $5 minus
5, which is 0 times 0.7.
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So you're going to
get nothing back.
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So there's really no good reason
for you to do that.
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Then if you give them $10.
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What's going to happen?
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If you give them $10, 10
minus 5 is 5 times 0.7.
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You're going to get $3-- or
I should say EUR 3.50.
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3.5 Euros, you'll get.
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Now what happens if
you give them $30?
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Actually let me say 25.
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If you give him $25,
25 minus 5 is 20.
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20 times 0.7 is $14.
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I'll do one more value.
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Let's say you gave them $55.
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This makes the math easy
because then you
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subtract that 5 out.
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55 minus 5 is 50 times
0.7 is $35.
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Is that right?
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Yep, that's right.
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You'll get EUR 35
I should say.
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These are all Euros.
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I keep wanting to say dollars.
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Let's plot this.
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All of these values are
positive, so I only have to
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draw the first quadrant here.
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And so the dollars-- let's go
in increments of 5, 10, 15,
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20, 25, 30, 35, 40,
45, 50, 55.
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I made my x-axis a little
shorter than I needed to.
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All the way up to 55.
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And then the y-axis.
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I'll go in increments of 5.
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So that's 5, 10, 15,
20, 25, 30, 35.
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Well that's a little bit too
much of an increment.
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35.
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Now let's plot these points.
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I give them $5.
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I get EUR 0.
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This right here is Euros.
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This is the dollars.
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The dollars is the independent
variable and we figure out the
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Euros from it.
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Or the Euros I get
is dependent on
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the dollars I get.
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If I give $10, I get EUR 3.50.
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3.50-- it's hard to read.
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Maybe 3.50 would be right
around there.
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If I give $25, I get EUR 14.
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25, 14 is right about there.
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Obviously, I'm hand drawing it,
so it's not going to be
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quite exact.
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If I get $55, I get EUR 35.
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So 55, 35 right there.
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If I were to connect to the
dots, I should get something
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that looks pretty
close to a line.
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If I did it-- if I was
a computer, it would
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be exactly a line.
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That looks pretty good.
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Then we could eyeball what
they asked us to do.
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Use your graph to determine how
many Euros you would get
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if you give the office $50.
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This is 50 right here.
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So you go bam, bam, bam, bam,
bam, bam, bam, bam.
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I'm at the graph.
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Then you go all the way--
actually I drew that last
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point on the graph a little
bit incorrectly.
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Let me.
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35 is right here.
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Let me redraw that point.
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35 is right there roughly.
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So 55, 35 is right there.
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So let me redraw my line.
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It will look-- I lost 25.
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25, 14 is right there.
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So my graph looks something
like that.
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That's my best attempt.
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Now let's answer the question.
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We give them $50 right there.
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You go up, up, up, up,
up, up, up. $50.
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The person is going to get.
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You go all the way to
the left-hand side.
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That's right about 31.50.
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We figured out exactly
using the formula.
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But you can see, you can eyeball
it from the graph and
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figure out any amount
of dollars.
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If you give them $20, you're
going to go all
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the way over here.
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You'll figure out that it should
be-- well $20 should be
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about 7.50.
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The imprecision in my graph-- in
my drawing the graph makes
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it a little bit less exact.
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When you say 20 minus 5 is 15.
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15 times-- actually it'll
be a little over
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$10, which is right.
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It's right over there.
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If you put $20 in there,
20 minus 5 is 15.
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15 times 0.7 is $10.50,
which is right there.
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So you can look at any point
in the graph and figure out
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how many Euros you'll get.
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Let's do this one where
we'll do a little bit
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of reading a graph.
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The graph-- I think it said
use the graph below.
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Oh, the graph below shows
a conversion chart for
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converting between weight
in kilograms
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and weight in pounds.
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Use it to convert the following
measurements.
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We have kilograms here
and pounds here.
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So they want 4 kilograms into
weight into pounds.
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So if we look at this
right here, 4
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kilograms is right there.
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We just follow where
the graph is.
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So 4 kilograms into pounds, it
looks like, I don't know, a
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little bit under 9 pounds.
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So a little bit less than--
so almost, I'll
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write almost 9 pounds.
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You can't exactly see.
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It's a little less than
9 pounds right there.
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4 kilograms. Now 9 kilograms.
We go over here.
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9 kilograms. Go all
the way up.
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That looks like almost
exactly 20 pounds.
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Here they say 12 pounds into
weight in kilograms. Actually
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kilograms is mass, but I
won't get particular.
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So 12 pounds.
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Go over here.
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Pounds.
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12 pounds in kilograms
looks like 5 1/2.
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Approximately 5 1/2.
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And then 17 pounds
to kilograms.
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So 17 is right there.
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17 pounds to kilograms looks
right about 7 1/2 kilograms.
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Anyway, hopefully that these
examples made you a little bit
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more comfortable with graphing
equations and reading graphs
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of equations.
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I'll see you in the
next video.
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