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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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00:11:45,269 --> 00:11:47,819
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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