Beg Algebra: Parallel & Perpendicular Line Slopes

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SLOPE FOR PARALLEL AND PERPENDICULAR LINES
Okay, so now I want to think about what does it mean for two lines to be parallel. What does it mean for two lines to be perpendicular? Well, parallel lines are just lines that are parallel. But how can you actually detect that kind of parallelism if you were just given the equation for a line? Well, if you think about it, if two lines are parallel, they must have the exact same pitch. Pitch results in slope. So two lines that are parallel precisely when their slopes are equal.
So let’s take a look at a particular example here. Suppose I take a look at the line y = 2/3x + 1. How would the graph of that play out? Well, this would have a y intercept. This is in point-slope intercept form. So y intercept of 1, and then a slope of 2/3. That means I go 3 units over in the x direction--1, 2, 3, and two up--1, 2. So I just put the line in there that connects those two points. And notice that no matter where you are on this line right here, the next point--1, 2, 3, 1, 2--is on the line. So, there’s that line. Now, here’s a line that looks very similar. In fact, what is this line? This is the line y = 2/3x - 2, and so it has the same slope, but now the intercept is -2. So I go down to -2, so 1, 2, there’s -2, but the same exact slope, so my slope now is 1, 2, 3, 1, 2. And so what I see here is a line that looks like this. As you can see, these lines are parallel. So parallel lines have the same slope and conversely, if two lines have the same slope, they must be parallel. Not a big deal. Parallel lines; same slope.
Perpendicular lines actually are much more complicated. I mean, not hard, but it’s just trickier to see what goes on. So let’s take a look at an example of perpendicular lines and see if we can detect a pattern. So the first line I want to take a look at is y = 3/4x. Now, notice that the y intercept is 0, so that means it goes through the origin. So it crosses the y-axis right at the origin. Okay, and what’s the slope? The slope is ¾. So I go 3 over--1, 2, 3, and 4 up--1, 2, 3, 4. And so now if I connect these people, that’s the graph of this.
Now, what I want us to take a look at is another graph. Now, this is a little weird looking, I admit. Let’s just take a look at this one together. Let’s take a look at y = -4/3x. What am I doing here? I just invert or flip the slope, that would give me 4/3. I also stuck on a negative sign. Let’s see what happens. Well, -4/3x, but still plus zero, so I still pass through the origin, and the slope is -4/3. That means I go 3 units over now in the x direction--1, 2, 3, but I go 4 units down, since I have -4--1, 2, 3, 4. So now I pass through this point, and if you do that, look what happens. These lines, in fact, look extremely perpendicular. In fact, maybe if you want to see it better, I’m just going to turn this around so you can try to see that perpendicularity. Look at that. In fact, I’ll put it right next to me. I’m going to nestle it right up against me. You can really see those lines are perpendicular. The interesting thing is, if you move it back what you see is that lines are perpendicular. What we learn is lines are perpendicular precisely when their slopes aren’t just flips of each other, but negative flips, or as we say in the business, “negative reciprocals.” So, in fact, if I see that the slope of a line is ¾, I know that the line that’s perpendicular of that will be the negative reciprocal of ¾, or in this case, -4/3. So that’s how you find perpendicularity and determine it in terms of slope.
Let’s take an actual look at some problems now or some questions that we are now empowered to answer just using that observation. So let’s try two of them. The first one is the following. Let’s find the line that passes through the point 1, 3, and is parallel--by the way, two lines like this is shorthand for parallel--parallel to 3x + 4y = -24. So this is the equation of a line, and I want you to find the equation of the line that’s parallel to this and goes through this point. How would you do it? Well, I need to first of all find the slope of the line, because I already have a point that I know is on the line, and now I’ve got to find the slope. Well, how does the slope of our line compare to the slope of this parallel line? Parallel line slopes are equal. So all I’ve got to do is find the slope of this line, and I know that will be the slope of our line. So how do you find the slope of this? I’ll just solve it for y. If I solve this for y I’d bring the 3x to the other side and I see 4y = -3x - 24. I just subtracted the 3x to both sides.
Now, if I divide through by the 4, I see y = -3x - 24, all divided by 4, and I could write that, if I wanted to, as y = -3/4x--I’m just breaking up this big fraction--24/4. The 24/4 I don’t care about, but I know that if I have it in this form, this is the slope-intercept form, and so in fact, I immediately know I can read off the slope of the line--it’s -3/4, and that it’s, -3/4. So if -that’s the slope, then what’s the equation of our line? I know a point, I know slope. I’m going to use point-slope form, which just says y minus the y value, so y -3, equals the slope, -¾, times x minus the x value, and our x value is 1. There’s the equation of the line. That is the line that passes through the point 1, 3, and is parallel to this line, and why is it parallel, because it has the same slope, and I found that by writing this out in this form. You see how neat that is?
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Let’s try one more question of the sort. Let’s find the equation of the line that passes through 1, 2 and is perpendicular--by the way, sometimes that’s denoted with a little angle like this--that’s perpendicular to x + y = 4. So what’s the equation of that line? I have a point; I need the slope. How do I find the slope? Well, I solve this for y, so I subtract x from both sides, and if I do that, I see in front of the x a negative 1. So the slope here of this line equals -1. Now, technically, that’s sort of -1 divided by 1, if you wish. Now, what do I do? Well, to find the slope of--so the slope of the perpendicular line--what do you call the negative reciprocal. So, I take negative and then I flip this, which equals - - 1, which equals 1. So the slope of our perpendicular line is going to have slope 1, so that’s m. I know it passes through the point 1, 2, so I’m going to again use point-slope form. You can see how handy this is. This is why this is my favorite. y minus the y value, so -2, equals the slope, which is just 1, times x minus the x value. So that’s what it looks like. I could rewrite that a little bit. I could say, well, y - 2 = x -1. That’s a fine way of giving the answer. Or if you wanted to write it in slope-intercept form, just add 2 to both sides and I would see y = x + 1. Either way is correct. This is the line that is perpendicular to this line and passes through the point 1, 2. It passes through the point 1, 2 because I put that point in, and it’s perpendicular to this line because the slope is the negative reciprocal of the slope of this line. Neat! All right. See if line are parallel or perpendicular and have some fun.