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Okay, so far we've covered quite a few things about graphing lines.
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Basically you've done this by plotting points.
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So for instance if you're asked to graph the line 2x minus 3y equals 6, the method we've
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used is to solve for why make a table of order pillars, plot them and draw the line.
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So for this problem, let's quickly do that. We'd have you could put this on pause and
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then check and see if you did it the same way I did.
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So I've got to subtract 2x from both sides. So I'm negative 3y equals negative 2x plus
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6 divided both sides by negative 3.
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So I have positive 2 thirds minus 2 and then we could pick any values for y. I'm sorry,
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for x. I'm going to choose multiples of 3 to avoid fractions. So I'm going to put a negative
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3, 0 and positive 3, 2 thirds times negative 3 minus 2. The 3's cancel I get negative 2
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minus 2. That's going to give me the order pair of negative 3 and negative 4. 2 thirds
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times 0 minus 2 is going to give me negative 2 and 2 thirds times 3. The 3's are going
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to cancel. So 2 minus 2 is 0. So using this method we would have negative 3, negative 4,
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0, negative 2 and 3, 0. Perfectly good method. You can always use this method if you can
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solve for y. The only time you can't solve for y if it's something like x equals 3. Remember
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and then you just put a bunch of order pairs where x is always 3 and you'll get that vertical
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line. Every other line that has a y in it, you're able to solve for y and plot points
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like this. Okay. So now what we're going to do is graph this line using a different
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method which actually is a little bit easier. Okay. So a second method for graphing equations
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of this form is to plot the x and y intercepts. Now the best time to use this method plotting
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by using x and y intercepts is when it's not solved for y. Okay. So for this problem here,
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2x minus 3y equals 6, it's not solved for y. So instead what we're going to do is notice
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that well on the graph, on the graph, on the x axis, y is always 0. You never go up
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or down. So if you want to find a point on the x axis, you know that the y value is always
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0. So every single ordered pair on the x axis is something 0. Every ordered pair on the
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x axis. On the y axis, you don't go left or right at all. x is always 0. So here's
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what we have. Okay. Since every ordered pair on the x axis has the form something 0,
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note y is always 0. This is how we're going to find the x intercept. The x intercept
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means where does the line cross or go through the x axis. We're going to plug in y equals
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0. So we have this equation to get the x intercept of 2x minus 3y equals 0, let y equals 0.
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So we're going to let y equals 0. So let's do that. We're going to plug in 0 for y to
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get the x intercept. So let's go over here and plug in 0 for y. That just says 2x is
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3. Now we know it goes through the point 3 0. Now to get the y intercept, it's the same
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idea. The y intercept of 2x minus 3y equals 6. If it's on the y axis, x is always 0. So
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we're going to let x equals 0. Okay. So we put in 2 times 0 minus 3y equals 6. Be careful
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if you're negative sign. You divide by negative 3y is negative 2. That's the value when x
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was 0, y was negative 2. Alright, so what do we get? The x intercept was 3 0 and the y
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intercept was 0 negative 2. So let's see what this graph looks like. Let's remember the
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equation. It was 2x minus 3y is 6. Alright, so let's plot the x intercept. That means
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it's on the x axis, 3 0. And the y intercepts on the y axis, 0 negative 2. Here's where
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you have to be really careful. The graph line, use a straight edge and kind of mark where
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it seems to be going through other points because that's how we're going to check it.
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So we've got that, we have 2 points, 3 0 in 0 negative 2. And you want to make sure
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the line's really going through the correct point. So this point over here, what is this?
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This looks like 6 2. That's the point 6 2. Is 6 2 a solution? Well, let's see, 2 times
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6 minus 3 times 2. Does that equal 6? Remember how we do that? We plug in 6 for x, 2 for y,
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12 minus 6 equals 6. Yep. So this looks like the correct line. 2x minus 3y equals 6. And
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so to plot the x and y intercept, to get the x intercept, you put in 0 for the y. To
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get the y intercept, you plug 0 in for the x. Okay, so if you have an equation of a line
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to find the x intercept, let y equals 0. You're going to get an order pair. And then to find
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the y intercept, let x equals 0 get a different order pair. Graph both points and do a checkpoint.
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By a checkpoint, my checkpoint was 6 2 up here. I have to see, make sure that another
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point that it went through was really a solution to the equation. Now, this only works if
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the x and y intercept are different. If it goes through the origin, that's only one point
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you're never going to get a line. So this isn't going to work unless it doesn't go through
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the origin. Alright, so let's do another example. Alright, so how would we find the x intercept
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in the y intercept of the line 5x minus 3y equals 15? To get the x intercept, I have to
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put in 0 for y. So write an ordered pair and immediately put in 0 for y. And now we're
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going to take the equation 5x plus 3y equals 15, plug in 0 for y. Alright, now when you
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put in 0 for y, hopefully you'll notice that's just 5x plus 0. So 5x is 15, x will be
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3. So now we have our answer for the x intercept. Alright, y intercept. We're going to put in
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0 for x. So immediately you write an ordered pair putting in 0 for x. Again, we're going
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to write the equation of the line. Now if we put in 0 for x, we get just 3y equals 15.
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Now be careful, if this was a minus 3y, you would have negative 3y equals 15. But it's
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positive. So 3y equals 15, y is 5. So that goes in for the y value of the ordered pair.
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And now we also have our y intercept. Okay, keep in mind that when you're using this
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method, the x and y intercepts, it's just that you're getting two points and plotting
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them just like before, except for specifically the x intercept and the y intercept because
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those are easy computationally putting in 0 for x and then just putting in 0 for y. So
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let's graph these. We have 3, 0, and then we have 0, 5. And then I have to graph this
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very carefully. I think that's where it's going to go through. Alright, it looks like
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it's going to be this point 6, negative 5. Let's check it. You want to make sure that's
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really on the line. So here's my checkpoint. So we're going to put in 6 for x, negative
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5 for y. That's the question. 30 plus negative 15 equals 15, yes, it's true. The other
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way to do it is to just take the original equation and plug in any other number for x and
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solve for y and plot that third point. But you do want to have three points on the line
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to check that equation.