Cartesian Plane – Point-Slope Formula for Lines
Lines on the plane: This is the imaginative title, lines on the plane part one. The point of this is to demystify formulas for equations of line. The first one, which we’re going to talk about in part one, is what we call the point slope formula for a line. Y minus y nod = m times x minus x nod. The first one’s a little bit more natural to derive. What we’ll do for both of these is work up slowly. Lines On The Plane, Part I • y-yo = M(x-xo) <- Point-slope • y=Mx+b <- Slope-Intercept, Part 2
We begin with an x-y Cartesian plane, where (A) is the point with x-coordinate little a and y-coordinate little b and (B) is the point with x-coordinate c and y-coordinate d. A line segment can then be drawn from A to B, whose length represents the distance between A and B. The slope of the line segment between points A and B, which we denote AB with an arrow, is often represented by the letter m. The slope is defined as the difference between the y-coordinate of the second point minus the y-coordinate of the first point over -coordinate of the first point, taking away the *-coordinate of the first point, c-a. This is often seen as rise divided by run and can be understood by example. [Graph] a Let’s compute a few real-world numerical examples. Suppose that we take the point (1,2), and let’s take another point (3,3). Let’s draw the line segment between these two points and ask what is the slope of this line segment? The slope of a line segment is defined as rise divided by run, or in this case 3 minus 1 over 3 minus 3 2. That turns out to be one half. When we say that the slope of this line segment is one half, we mean that it’s the answer to the following question. If we start f at A and move one unit over. increasing our x coordinate by 2, how do we get to the point where we stay on the line but increase our y coordinate by 1.5? Since we want to stay on this line and increase our y coordinate by 1.5, we must rise 2 times 1/2 units in the y direction and so arrive at point B with coordinates (3, 3). A positive slope means that as you run in a distance in the x direction, you will rise up a distance in the y direction. A negative slope means that as 3 you run a distance in the direction, you will fall down a distance in the V direction. We’ll take the point C here. with coordinates -1.1. We’ll draw a line segment to the origin. This is 0,0, the origin. There’s a nice little segment. The slope of the line segment CO S 0- divided by 0 – (-1), careful about subtracting a negative, that’s – 1 over 1 is – 1 a So there is a negative slope. That makes sense if we’re going to run one unit in the x direction we have to rise – 1 or fall one unit down the Y axis, down here. That’s the idea. [Graph] Now let’s discuss the slope of a line segment, which is defined as the change in y divided by the change in x for a given pair of points on the line. Let’s take our original example: The line segment between two points (2,1) and (3,2) has a slope equal to 1 because it rises one unit for each unit that it moves horizontally. However, we can see that if we extend this line infinitely in both directions – going down like this and calling it little I -we can see that any point on that line has a slope equal to 1 as well. This makes sense because all points along a single line have the same slope; they are essentially all just different sections of one long line. In other words, 1 has to equal the difference of y -1 divided by x minus 2. The difference in the rise from 2,1 to x,y divided by the run. Now let’s rewrite that as y- 1 = 1 index- 2. This is actually a really profound statement. Because xy was arbitrary with any point on the line. This has to be true. So in other words, the line is an exclusive club which is defined by this formula as a set, the line is a set of O.xy in the Cartesian plane. Such that the following relationship in x and y values is true. Y-1 is equal to 1 times X-2. Let us check if that works. We know that three times two equals six, so if we plug two into this formula and three into x, we get: 2- 1 is 6 equal to 6 minus 3. If we work this out, we see that 1 equals 1 times 1, which is true. So any point on the line has to check whether this statement is true. For example, the point (5,1) is not on our line because if we plug in five for x and one for y, we get an incorrect statement: 5-1 is not equal to six minus three. [Graph] Let’s write down formally what we just concluded. This is called the point-slope formula of a line. If a line, 1, has slope M and passes through any point (x0,yO), then its equation is: y – y0 = m(x – x0).
Point-Slope Formula If a line, I, has slope M and if (xo, yo), is any point on the line. Although we don’t tell you what it is, is any point on the line, then I has the equation. Y- yo = M (x- x0)
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