Slope, the single most important idea to understand about lines in the x-y plane is the idea of a slope. Slope is, roughly, a measure of how steep a line is. Different lines are at different angles, and slope is a way of talking about these different slants. One of the most common definitions of slope is rise over run.

What does this mean? Suppose we want to find the slope between two points. The run is the horizontal separation from the left to the right. This is always positive from left to right. The rise is the vertical separation between the two points as we go from left to right.

If the point on the right is higher, then the rise is positive. If the, if lower, it's negative. Suppose we want to find the slope between (2,2) and (6,5). We draw or imagine a little slope triangle. So, this slope triangle allows us to see there's a horizontal separation and there's a vertical separation.

The horizontal separation, the run, is four, and the vertical separation is three. And we can see this just from the legs of the slope triangle. So the rise of 3 over the run of 4, rise over run. So this line has a slope of three-fourths, 3 over 4. Suppose we want to find the slope between (-4, 2) and (5, -1).

So we draw or imagine this slope triangle, and notice that the vertical leg is three. In effect, since it's higher on the left, it's gonna be a drop, so we have a quote, unquote, rise of negative three, and then the run is that long horizontal leg, that has a length of nine. So the slope, rise over run, negative 3 over 9, and of course we can simplify this fraction to negative one-third.

And that is the slope, a slope of negative one-third. If you are given the numerical coordinates of two points, I would suggest using the slope triangle to figure out the slope. In other words, thinking about it visually with the slope triangle. Sometimes you are given an algebraic, you are given algebraic information and you have to use a formula for the slope, but do not make the formula your default.

This is one of many examples in which over-reliance on the formula produces shallow mathematical understanding. You actually understand it much more deeply when you're thinking about it visually. After all, the x-y plane is visual. It's something you see.

Having warned you about the perils of over-reliance on the formula, I will give you the formula. Suppose we have two general points, and let's say the one on the left is (x1,y1). The one on the right is(x2,y2). It actually doesn't really matter which one's on the right or which one's on the left.

The run, is gonna be x2 minus x1. The rise is gonna be y2 minus y1. And the slope is gonna be rise over run. So we get this formula, y2 minus y1 over x2 minus x1. So yes that formula is handy in certain situations, but if you just think about it in terms of rise and run, you'll have a much deeper understanding.

So far we have been talking about the slopes of two individual points. A line has a slope, and the slope of a line is the slope between any two points on the line. So that's a big idea. We could pick any two points, any two points at all of the whole infinity of points on that line and there would, we'd always find the same slope between those two points.

If a line has a slope of m equals one-half, then we will pick the two points on the line, and the ratio of rise over run will always simplify to one-half. So fact, if you think about the different slope triangles it would draw on this line, they would all be similar triangles, so they would all have the same ratio. Let's think about a slope of m equals 2 This could mean that we are moving to the right one unit, and up two units.

It could also mean that we move to the right k units, and then we'd have to move up 2k units. It could mean that we move to the left one unit and down two units. Similarly, we could move to the left k units and down 2k units. So it might just be a step of one and then a step of two, one over and two up, but it could be any larger amount in that ratio.

We could go five over and ten up or 15 over and 30 up, or something like that. So for example, if (-3, -1) is a point on a line with a slope of m equals 2, then we could find other points. For example, if we went over one and up two, so over one from negative three would be negative two. Up two from negative one would be positive one, so that would be the point (-2,1).

Or, we could start at (-3,-1) and we could go to the left one and down 2. So, if we start at negative three and we go to the left one, we get to negative four. If we start at negative one and we go down two, we get negative three. So (-4,-3) is also a point on this line. So given a point and a slope, you should be able to find other points.

Let's think about a slope of negative two-thirds, a fractional slope. This certainly could mean a run of three and a rise of negative two. Or in other words, to the right three units and then down two units. That's one thing it could mean. It could mean to the right 3k units and down 2k units, again as long as we're going in the same proportion.

We could go to the right say 30 units and down 20 units, something like that. Or, we might think in terms of fractions. We might move over one unit and then go down two-thirds of a unit. And then, we'd think about fractional movements downward. We could also go to the left. We could to the left three units and up two units.

Or, to the left 3k units and up 2k units. Or to the left one unit, and up two-thirds of a unit. So there's lots of different things that a slope of negative two-thirds can mean. Here's a practice problem. Pause the video and then we'll talk about this.

Okay. If a line goes through the point (2,-1) and it has a slope of m equals five-thirds, find all the points (a,b) on the line where a and b are integers whose absolute values are less than or equal to 10. So we're gonna start at that point (2,1) and we're gonna move to the right three and up five.

Now notice if we moved any fraction over, then we, we, yes, we'd land on a point on the line, but it wouldn't be both, both the x and the y coordinates integers. So we have to move three to the right and five up. And so if we start at (-2, 1), three to the right of negative two would put us at five, and five up from negative one would put us at four. That's one point.

Then we can move three to the right and five up again. Add three to the x coordinate add five to the y coordinate. That brings us to the point x naught (8,9). If we do it again, then we get points whose absolute values are greater than ten, so that doesn't count. We've found two points that work, and then we've ran up to a point that doesn't work.

Well, we can also move to the left. That is left three and down five. So we start at (2,-1). We'll move, we'll subtract three from the x-coordinate and move down to negative one, subtract five from the y-coordinate, move down to negative six. So (-1, -6) that's a point on the line.

If we do this again, we're gonna go down to y equals negative 11. Again negative 11 has an absolute value greater than ten so this doesn't count. So we've found three points that satisfy this condition that are on the line in addition to the point given in the prompt. Notice that if a line has a slope of one, then rise equals run, and the slope triangle is a 45-45-90 triangle.

That's a really big idea. You should know that lines with slopes of, of one or negative one make 45 degree angles with the each one of the axes. That's really important. You don't have to worry about the exact angles formed by any other slanted lines. Simply notice that if the slope is greater than positive one or less than negative one, then you're gonna have an angle greater than 45 degrees.

And, of course, that's gonna be steeper than any road that you typically would walk on or drive on. What is true of the slopes of, if two lines are parallel? If the lines are parallel, they must rise in sync, with the same rise for the same run. In other words, parallel lines have equal slopes.

That's a big idea. Parallel lines always have equal slopes. What is true if the slopes of two lines are perpendicular? First of all, if one line goes up, then the other must go down. In other words, the slopes must have opposite signs. If one is positive, the other must be negative.

The opposite signs is part of the answer, but we need to think of the numerical value of the slope. Think about what happens when we rotate the slope triangle by 90 degrees. So there we have an original slope triangle and then it's rotated by 90 degrees, so what was the original rise becomes the run, and what was originally the run becomes the rise.

So the rise and the run switch places. In other words, the numerator and the denominator of the slope fraction have switched places. Well, that's a reciprocal, that's what we call a reciprocal, when you switch the place of the numerator and the denominator. The slopes of perpendicular lines are opposite-signed reciprocals.

In other words, if the slope of the original line is p over q, then the slope of the perpendicular line, we make the positive negative, and we flip the fraction over. If the original slope is positive one-half, the perpendicular slope is negative two. Slope is rise over run.

We find the slope between two points with a slope triangle, or with the slope formula. And again, I would urge you to think more in terms of the slope triangle and not rely too heavily on the slope formula. The slope of a line is the slope between any two points on the line. Lines with slopes of positive or negative one make 45-degree angles with the axes.

Parallel lines have equal slopes. And, perpendicular lines have slopes that are opposite-signed reciprocals of one another