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Okay, we're going to go over some more examples, graphing lines.
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There's two methods we've gone over so far using a T chart where you pick a value of x
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and put it in the equation and solve for y to get an order pair and you find a few ordered
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pairs and graph a line.
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And the other method we did was the x and y intercept method where you find the x intercept
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and the y method.
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The x and y intercept method works for most lines but not if it's going to the origin
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or if it's a vertical or horizontal line.
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So let's start with an example.
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5x minus 3y equals 15.
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Now since this is not solved for y, let's try the x and y intercept method.
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Okay, so to get the x intercept,
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remember you are going to put in 0 for y.
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So your order pair, your halfway there, put in 0 for y into the equation.
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So we have 5x minus 3 times 0 equals 15.
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Now, you don't necessarily have to write all this out.
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You might go to look at it and realize when you put in 0 for y, you'll simply get 5x equals 15.
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And then if you divide both sides by 5, you get x equals 3.
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So 3 is 0 is the x intercept.
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So let's get the y intercept now.
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The y intercept, remember we put 0 in for x because everywhere on the y-axis,
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you're not moving left or right at all.
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So for putting in 0 for x, we have 5 times 0
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minus 3y equals 15, which says negative 3y equals 15.
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And if you divide by negative 3, be careful we get negative 5.
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So we've got 0 negative 5.
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So here's a way of graphing the line by using the x and y intercepts.
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We find each of them, we'll plot the x and y intercept and then graph it.
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Alright, so you get out graph paper and let's graph this.
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We have 3, 0 will be right here and 0 negative 5 will be right here.
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And then we draw carefully the line going through those two points.
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And as long as we didn't make a mistake, we've got the line graphed for 5x minus 3y equals 0.
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So we use the x and y intercept method.
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Let's do this same problem using a t chart.
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So I'm going to go ahead and leave the graph we got by using the x and y intercept method.
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And to use the t chart, let's solve for y.
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So we have negative 3y equals negative 5x plus 15.
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Divide both sides by negative 3.
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And we have 5x minus 5.
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So we've got y equals 5x minus 5.
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And now we would plug in some numbers for x and then compute 5x minus 5 to get a value of y.
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Now since we're doing 5x, let's pick values of x that are multiples of 3.
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So negative 3, 0 and 3 might be a good choice.
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So I have 5x times negative 3 minus 5.
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Alright, so what do we get there?
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The 3's cancel of 5x negative 1.
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So negative 5 minus 5 is negative 10.
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And then we're going to plug in 0.
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Let's see, can you do that in your head?
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If you put in 0 up here in the equation, you're going to get y equals negative 5.
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So when you put in 0, you're going to get negative 5.
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I'm not going to show that computation.
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And then 5x times 3 minus 5.
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The 3's will cancel on 5 minus 5 is 0.
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So let's do this with some green marker.
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I'm going to plot these points.
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Well, negative 3, negative 10, negative 3, that's way off the chart.
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It's somewhere down here.
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I could estimate it.
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Let's try that.
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1, 2, 3, then this goes at, there's negative 6, right?
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So that's negative 8, 9, 10.
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So it looks like it's a bell right here.
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Yeah, it looks like it's going to be on that line.
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I'm approximating, of course.
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Let's do 0, negative 5, hey.
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Same thing as I had when I did the other way, which was the x and y intercept.
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And 3, 0, hmm.
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That also just happened to be the x intercept.
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I don't know if I, the 0 negative 5 was the y intercept, sorry.
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And so you can see that this is going to be the same line.
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I'm using a t chart coming up with the same graph.
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Now, which do you think is easier?
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I think it was easier to do the x and y intercepts because it was simple to plug
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in a value for, plug in 0 for x and solve for y and plug in them, plug in 0 for y and
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solve for x.
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So I would use the x and y intercept method if your original equation here is not
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solved for y, usually it's easier to use the x and y intercept method.
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Keep in mind though, you might get a fraction for the x or y intercept, but that's
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okay.
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Fractions are numbers, too.
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Here's another problem, 3x plus 2y equals 0.
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So this is not solved for y, so let's try using the x and y intercept method.
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So we try getting the x intercept by putting in 0 for y.
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Remember?
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So we put in 0 for y and hopefully you see that that's going to give you 3x plus 0 or
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3x equals 0.
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Okay, so if you divide by 3, we're going to get x equals 0.
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So I got 0, 0.
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You know what?
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If 0, 0 is one of the points on the line, the origin is on the line and you're not going
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to get a different x and y intercept.
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You've got the same x intercept as you do for the y intercept.
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So if you plugged in 0 for the other value, so in other words, if I was trying to get
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a y intercept, look what would happen.
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You had to put in 0 for x and got in 2y equals 0 and when you divide by 2, you get the
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same point.
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So there's a problem here.
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I wasn't able to get two different points, so how can I draw a line if I only have one
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point?
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So in this particular example, notice the constant.
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I've got 0 over here.
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If 0, 0 is a point on the line, the x and y intercept method is not going to work.
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So you know what?
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We're going to have to use a t chart.
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That's one of the methods we could use, I better say that.
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It's just that you do not have to use.
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I mean the x and y intercept method will not work.
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All right.
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So I at least have the 0, 0.
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So there's nothing wrong with leaving that as one of my order pairs, but let's go ahead
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and try solving for y.
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So if I solve for y, I'm going to subtract 3x from both sides and then divide by 2 so
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that y equals negative 3,0x.
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So let's plug in some values for x into negative 3,0x and find the values of y.
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Now remember, we already know 0, 0 is on the line, so I don't need to do that again.
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Let's just pick two more points.
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How about something where 2 is going to cancel in for x?
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So how about negative 2 and then positive 2?
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So we've got negative 3 halves times the negative 2 and then we'll do negative 3 halves times
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the positive 2.
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All right.
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So for the first one, the 2's cancel and negative, negative times negative is going to be positive.
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So my answer is going to be 3.
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When I put in negative 3 halves times 2, the 2's cancel and so I get negative 3.
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This gives me two other ordered pairs, negative 2, 3 and 2, negative 3.
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And now I've got, well, as long as you have two points, it's going to work, but it's
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even better if you can get three points on the graph.
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So I've got three points on the graph because I had already figured out 0, 0 was on it
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and there's my line.
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So here's the deal.
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If it's solved for y, it's easier to use a t chart than the x and y intercepts.
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So if I give you the equation y equals negative 3 halves, this would probably be the simplest
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way to do it of these two methods.
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Or if I gave you y equals 3x plus 1 or anything where it was solved for y.
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If it's not solved for y, try to use the x and y intercept method and if it doesn't
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work because it goes through 0, 0, then you can solve for y and proceed.
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So let's see.
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Here's some, here's our four equations and let's say you had to graph them.
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Which one do you think would be easier to use to graph these equations?
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Think about that.
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The t chart where you just find some ordered pairs or the x and y intercept method.
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Go ahead and figure that out and put it on pods and then come back.
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All right, first one, it's solved for y.
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So you know what, t chart may as well.
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Just plug in some, any number you want for x and solve for y and you've got and get
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a few ordered pairs.
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Second one, it's not solved for y.
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So you would probably use the x and y intercept method.
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It's pretty easy.
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If you plug in 0 for x, find the y value.
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It'll give you the y intercept and then plug in 0 for y and get the x value.
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That'll get the x intercept.
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How about this one?
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x minus 3y equals 0.
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At first glance, you might think, well, I would try to use the x and y intercept method.
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But if you try that, you'll find out that 0, 0 is an intercept.
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So the x and y intercept method won't work here and you'll have to end up using the
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t chart.
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And the last one, it's solved for y.
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So mays will use the t chart.
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But this is just what I think might be easier.
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The x and y intercept method will work on all of these except the third one.