WEBVTT
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Let's talk about lines.
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Two of the most important skills regarding
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lines are graphing a line and writing the
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equation of a line.
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This example will have us do both.
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We'd like to sketch the graph of a line with
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slope negative 2/3 passing through the point
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1, negative 2.
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We'll start by plotting that point 1,
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negative 2, and then we'll use the slope to
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determine a second point on the line.
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So our slope here is negative 2/3.
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Remember that slope can be thought of as the
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rise over the run, or the change in y over the change in x.
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We're going to use this to plot that second point.
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So, negative 2/3, and we're going to put the
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negative sign with the 2, so we have a negative rise.
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We're going to go from our starting point
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down two units, and then the run is positive
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3, so we're going to then go to the right three units.
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This will give us the second point on our line.
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That is enough to graph the line, but let's
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go ahead and find a third point.
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Suppose that I like to put that negative
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sign in the denominator with the 3.
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So, now my rise is 2, and the run is negative
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3, so I'm going to go to the left three units.
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What you'll notice is that all of those
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points are in a straight line,
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either of two of those would do.
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Now, let's talk about writing the equation of this line.
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We're going to start with point-slope form,
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and in point-slope form, the m stands for
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the slope as usual, and then x1, y1 are
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going to be the coordinates of any point on the line.
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We can use the one we're given 1, negative 2, as that point.
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So, let's substitute, y1 is negative 2, so
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we have y minus a negative 2, the slope is
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negative 2/3, and x1 is 1.
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I'm going to go ahead and distribute on the right-hand side.
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Then let's think about how our answer should look.
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Standard form is A x plus B y equals C.
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So, we need to start out by getting the x
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and y terms on the left and put everything
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else on the right-hand side.
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Let's add 2/3 x to both sides.
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I want to subtract this 2.
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So, we have to do 2/3 minus 2.
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Okay, that's 2/3 minus 6/3 or
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negative 4/3, and we're very close.
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It's customary in standard form for the
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coefficients to be integers when possible.
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We can do that here by multiplying
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everything, each term all the way across, by 3.
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So, that gives me 2 x plus 3 y equals negative 4.
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This is the equation of the line in standard form.
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Now, let's look at another example.
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In this example, we're also looking to
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sketch the graph of a line and write its equation.
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Here, we look at two special types of lines however.
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We want to sketch the graph and write an
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equation for the line that passes through
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the point negative 2, 3 and has an undefined slope.
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Okay, so let me start by plotting the point negative 2, 3.
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A line with an undefined slope is a vertical
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line, so we want to draw the graph of a
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vertical line that passes through the point negative 2, 3.
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Notice that it has an x-intercept, this line, at negative 2.
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So, the equation of this vertical line is
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just x equals negative 2.
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In this second part, we want to sketch the
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graph and write an equation for the line
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passing through the same point, negative 2, 3.
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Now, this one has a slope equal to 0.
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So, again, I plot my point, negative 2, positive 3.
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Now, a line with a slope equal to 0 is a horizontal line.
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So, I'm going to draw my horizontal line through this point.
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And then, notice that this one crosses the y-axis at 3.
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That's its y-intercept. So, the equation of the horizontal line
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through that point is y equals 3.
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Let's look at a few more examples.
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This example asks us to write the
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slope-intercept form of the equation of a line given two points.
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To write the equation of a line, we always
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want to start out with point-slope form.
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In point slope form, we need to know the
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slope, m, and the coordinates of one point
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on the line to call x1 and y1.
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In our example, we're given two different points.
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So the first thing we need to do is find the slope of the line.
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Let's do that here.
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The slope of a line is the change in y over the change in x.
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Given two points on the line, x1, y1 and x2, y2,
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we can write the slope formula here and solve to find the slope.
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Alright, so let's just make a choice.
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I'm going to call this one x1, y1 and this one x2, y2.
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y2 is negative 5 minus y1 which is 1 over x2
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which is 2 minus x1 so I get a plus 2.
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Okay, so this is negative 6/4 which is equal to negative 3/2.
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Let me point out to you that I could have
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chosen this to be x2, y2 and this to be x1, y1,
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and I should have ultimately come up with the same slope.
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Now I'm ready to use my point-slope form from before.
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Alright, so I've called this y1, I'll go ahead and use that: y1 is 1.
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The slope we found to be negative 3/2, and
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x1 is negative 2, so that makes that x plus 2.
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Alright, so I've written an equation for the line.
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I want it to be in slope-intercept form.
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This means that I want to solve for y.
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First of all, let's distribute this.
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So that's negative 3/2 x.
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Negative 3/2 times positive 2 is going to
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be, give me a minus 3 here, and I want to solve for y.
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So, add 1 to both sides. That would give me
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a minus 2, and this is in the form y equals m x plus b.
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This is the slope, this is the y-intercept,
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and that's your slope-intercept form.
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And, again, let me point out that I could
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have chosen this point to be the x2 and y2
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and this one to be x1 and y1 and using the
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other point, I should have still gotten the same equation.
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In this example, we are again asked to write
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the equation of a line, but as you'll see,
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we're given slightly different information.
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We'd like to write the equation in standard
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form of the line passing through the point
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negative 3, 2 and parallel to 2 x minus 4 y equals 5.
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We know that to write the equation of a
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line, we should almost always start with point-slope form.
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And to use point-slope form, we need to know
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the slope of our line, and we need to know
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the coordinates of one point, x1, y1.
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So we are given that point, and we can use this information.
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We need to figure out the slope of our line.
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We are given something that will help.
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Our line should be parallel to 2 x minus 4 y
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equals 5, and there's a relationship between the slopes of parallel lines.
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Let's see if we can find the slope of the line.
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What's a good way to do that?
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One is to write it is in slope intercept
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form, that means we want to solve for y and
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have y equals m x plus b, so we can just pick out that slope.
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Let's subtract 2 x from both sides and
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then divide by negative 4.
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So we have negative 2 over negative 4 is
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positive 1/2 and then 5 over negative 4,
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actually going to make this minus 5/4.
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Okay, so the slope of this line is 1/2.
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What's the relationship between the slopes of parallel lines?
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Right, they're equal.
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So, if the slope of our line is 1/2, the
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slope of any line parallel to it is also ½.
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That's the last piece of information we
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needed, we can substitute into this equation
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and write our line in the appropriate form.
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y1 is two, our slope is 1/2, and we have x minus a negative 3.
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Let's distribute the 1/2.
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Okay, go back and look, we wanted this in
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standard form, that's A x plus B y equals C,
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so we are going to subtract the 1/2 x from both sides.
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And we need to add 2 to the 3/2.
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Okay, add it to both sides, 2 plus 3/2,
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that's 4/2 plus 3/2, or 7/2.
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It's customary for these coefficients to be
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integers, it's also customary to write the
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coefficient of x as a positive number in standard form.
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We can accomplish both of those things by
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multiplying each term by negative 2.
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And this is the equation of our line written in standard form.
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In this example, we again want to write the
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equation of some lines, given some characteristics of those lines.
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Here we want the equation of a line passing
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through the point 3, 2 that is first
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parallel to y equals 5, so let's draw a picture of this.
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Here's the point 3, 2, and let's go ahead
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and graph the line y equals 5.
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Now this says that y is always 5, so for
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every point on the line the y-value, the y-coordinate, has to be 5.
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This must be a horizontal line, and of
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course it'll have to cross the y-axis at 5
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because every point must have a y-coordinate of 5.
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Okay, the line we're interested in is
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supposed to be parallel to y equals 5, so I
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know right away that the line I want needs
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to also be horizontal, but it has to pass through this point.
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Let's draw a horizontal line that passes through this point.
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What does the y-coordinate of every point on this line have to be?
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Two, that's right, so the equation of this line must be y equals 2.
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Let's look at the second part of the question.
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We again want the equation of a line passing
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through the point 3, 2, but this time we
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want our line to be perpendicular to y
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equals 5, so again let me plot the point 3,
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2, and graph the horizontal line y equals 5.
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Alright, the line we're interested in needs
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to be perpendicular to y equals 5, so yes,
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it has to be a vertical line.
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We want this vertical line to pass through the point 3, 2.
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Now, what's special about this line?
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Well for one thing, the x-coordinate of
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every point on the line has to be 3.
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And that's what's going to give us the equation of our line.
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Since the x-value is always 3 no matter the
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y-value, the equation of the line is x equals 3.
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Be sure to practice all of these skills
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regarding lines so that you are comfortable doing the problems.