# How We Finally Learned to Tell Lines Apart

## Метаданные

- **Канал:** polymathematic
- **YouTube:** https://www.youtube.com/watch?v=49hlcR7P0kM
- **Источник:** https://ekstraktznaniy.ru/video/26432

## Транскрипт

### Segment 1 (00:00 - 03:00) []

Consider a line. Actually, consider dozens of lines. For the primary tools of geometrical construction, the straight edge, that is the line, and the compass, a circle. For those purposes, any line is as good as any other line. What matters is what you can draw with them. Let's go back to just two lines for a second. The two lines pictured here are clearly not parallel. They intersect, but otherwise, there's no sense of place or orientation. You might be able to say something about the angle between them. If perhaps the lines were situated in a triangle, for example, but otherwise you find yourself very limited. But it's here that the coordinate plane knits together algebraic and geometric understanding. It gives us additional tools to describe and clarify the relationships. If we situate these two lines in the coordinate plane, there's so much more we can begin to say about them. We can identify particular points on the lines. We can use those points to describe where the lines start in some sense and how they proceed. We can describe that procession with slope, which is simply describing how the lines change up and down compared to how they change left to right. If we bring back a bunch of those original lines and we think about them now in terms of slope, we do get a distinction between some of them. We can sort them by positive slope or negative slope. Or if we do go back to just these two lines and we define their slopes and we write their equations in terms of that slope and where the line starts, which we more commonly call the y intercept, we get the origin of the dreaded slope intercept form. We can take those points from earlier -1, -5 for the blue line and 2, 1, and we can use a formula to compare the change in y to the change in x. This is what we call slope. We simply subtract the different coordinates. So -5, the first y-coordinate on the blue line, minus one, that second y-coordinate, which is going to give us a result of -6. And then we're going to compare that to the horizontal change. Again, going back to an x coordinate now, -1 minus the other x coordinate of 2, which is -3. The ratio -6 to -3 is 2. And so we would say the slope of this blue line is 2. For our red line, we might note points at -1. 5, 4. 5. And then another point here at 1, -3. And then using exactly the same process, subtracting y-coordinates, 4. 5 - -3 makes 7. 5. And -1. 5 - 1 makes -2. 5. We get a ratio of 3 over -1. As we simplify this, or more simply -3. Combining this with those locations that the lines started, which we also call their y intercepts, we get equations like y = 2x - 3 or y = -3x + 0. When Decart situated geometrical figures like lines and curves in the coordinate plane, he gave us an entirely new veilance with which to understand the figures. Equations can now be associated with figures. Certain curves correspond to certain kinds of equations.