Algebra

Slope-intercept form | Algebra I | Khan Academy

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There are many different ways by which you can represent a linear equation. For example, if you have the linear equation y is equal to 2x plus 3, this will be one way to present it. But I could portray it in countless ways. I could take 2x out on both sides and we get: minus 2x plus y is equal to 3. I can process it in ways I can get it as … I’m not going to do it right now, but here’s another way to write the same thing: y minus 5 is equal to 2 minus x 1. In fact, you can simplify that and you can get either this equation here, or this equation above. They are all equivalent, you can get from one the other with the help of logical algebraic operations. Yes, there are countless ways to represent a linear equation, but in this video I want to focus on this presentation, because it is a very useful representation of a linear equation. In the following videos we will see that this and that can also be useful, depending on what you’re looking for, but we will focus on that. He is often called equation for given angular coefficient and intersection point. Equation for given angular coefficient and intersection point. I hope in a few minutes to make it clear why it is called so. And before I explain it, let’s try to draw it. I will try to draw it, I will make a few points, x; y and I will choose several values \u200b\u200bfor x, in which we can easily calculate the values \u200b\u200bof y. Perhaps the easiest is if x is zero. If x is zero, then 2 is zero, this article disappears and you only stay with this article here, y is equal to three. And if we have to put it on the schedule … In fact, let me apply it. This is the y-axis, let’s do the x-axis, oh not so right as much as I would like. It looks pretty good. This is the x-axis and let’s put a few dashes here, x equals 1, x is equal to 2, x is equal to 3, this is y equal to 1, y equals 2, y equals 3 and obviously I can go on many more. This will be equal to minus 1, this will be x equal to minus 1, minus 2, minus 3, and so on. This point here is zero; three – here x is zero, y is three. Okay, the point that represents x equals zero and y equals three, is this, we are right here on the y-axis. If we have rights to go through it, that right too to contain this point, this will be the point of intersection with the y-axis. The reason why this species is called given the angular factor and the intersection point is that it is very easy to calculate the intersection point with the y-axis. The point of intersection with the y-axis here will be when – written in this form – when x is zero, and y is equal to three, will be this point. It is very easy to find the point of intersection with the y-axis of this type of equation. But you can tell if we have an equation of the kind at a given angular factor and intersection point, it is probably very easy to find the angular coefficient (slope). And this conclusion is very true! We’ll see about that in a few seconds. Let’s make a few more points here and I will continue to increase x by one. If we increase x by one, we can write, that this is delta x, the change to x, the Greek letter “delta”, this triangle is the Greek letter “delta”, it represents the change in a given quantity. The change in x here is one. We just increased x by one. And what it will be the corresponding change in y? What will be the change in y? Let’s see when x is equal to one, we have two at a time, plus three will be equal to five. The change in y will be two. Let’s do it again. Let’s increase x by 1. The change in x is equal to 1. When increasing from 1 we will go from x equal to 1 to x is equal to 2. What is the corresponding change in y? When x is equal to two, two by two is four, plus three is seven. The change in y is equal to two. We go from five … When x went from one to two, y goes from five to seven. So for every 1 that we increase x, y increases with 2. So for this linear equation the change in y on the change in x will always be, the change in y is two when the change in x is one or is equal to two, or we could say that the angular coefficient is equal to two. Let’s draw it to make sure that we understand it. When x is equal to one, y is equal to five. We’ll have to plot five on the chart up here. When x is equal to one, y is equal to … in fact it’s a little higher, let me delete a little. This will be… I will delete it a bit. Like this. That’s equal to four, and that’s equal to five. When x is one, y is equal to five, this is the point. The rights will look … you only need two points to define a line, the rights will seem let’s do it with this color here. The rights will look thus. Let’s see now … I didn’t draw to scale, but it will look similar. This is the right y is equal to 2x plus three. But we have already found that its slope is equal to two, when the change in x is one, the change in y is two. If the change in x was minus one, the change in y is minus two. You can see that this is the case if we go down from scratch with one and go to minus one. Then how much will y be? It’s two minus one minus two plus three is one. We see that the point -1; 1 is also from the rights. The angular coefficient, or the change of y over the change of x, if we move between two arbitrary points of this line, there will always be two. But where do we see the number two in this initial equation? You see two here. And when you write an equation in the form of a given angular factor and intersection point, where you are explicitly looking for y, y is equal to some constant of x in the first degree, plus some other constant, the second will be the intersection point with the y-axis. That is, it is a way to find the intersection of the y-axis, the point of intersection is the point at which the line intersects the y-axis. Then 2 will represent the angular coefficient (slope). And this is logical, because every time you increase x by one, you’re going to multiply that by two, and you will increase y by two. This is dating with the idea of \u200b\u200ban equation for given angular coefficient and intersection point, but for me this is the easiest way to find out what the graphics look like of something because if it was given to you another linear equation, for example y is equal to minus x plus two, you will immediately say, “Okay, the point of intersection with the y-axis will be 0; 2, so I will intersect the y-axis right at this point and then i have an angular factor … The odds here are minus one, so I have a slope of minus one. When we increase x by one, we will reduce y by one. You increase x by one, you decrease y by one. If I increase x by two, I will decrease y by two. Then the rights will look like this. Let’s see if I can draw it relatively correctly. It will look … I think I can do it a little better than that. This is because my graphics are made by hand. It’s not perfect, but I think you understand what are you talking about. It will look similar. We understand that from the equation given the angular coefficient and the intersection point it is very easy to find the intersection with the y-axis and the angular coefficient. The slope here is that’s minus one, and the intersection point with the y-axis is the point 0; 2, very easy to find, because here you have all the information.

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