How to solve the equation of a straight line through two points?

Mathematics is not a boring science, as it seems at times. It has a lot of interesting, though sometimes incomprehensible for those who do not want to understand it. Today, it will be about one of the most common and simple topics in mathematics, or rather that of its area that is on the brink of algebra and geometry. Let's talk about direct and their equations. It would seem that this is a boring school subject that promises nothing interesting and new. However, this is not so, and in this article we will try to prove to you our point of view. Before turning to the most interesting and describing the equation of a straight line through two points, we turn to the history of all these measurements, and then find out why this was all necessary and why now also the knowledge of the subsequent formulas will not interfere.

History

Even in ancient times, mathematicians were fond of geometric constructions and all sorts of graphs. It is difficult to say today who first came up with the equation of a straight line through two points. But we can assume that this man was Euclid - an ancient Greek scholar and philosopher. It was he who, in his treatise "The Beginnings", originated the basis of future Euclidean geometry. Now this section of mathematics is considered the foundation of the geometric representation of the world and is taught in the school. But it should be said that Euclidean geometry operates only at the macro level in our three-dimensional measurement. If we consider the cosmos, it is not always possible to represent with it all those phenomena that occur there.

After Euclid there were other scientists. And they perfected and comprehended what he discovered and wrote. In the end, it turned out a stable area of geometry, in which everything is still unshakable. And for millennia it has been proved that the equation of a straight line through two points is very easy to compile. But before we begin to explain how to do this, we will discuss a bit of theory.

Theory

A straight line is an infinite in both directions segment, which can be divided into an infinite number of segments of any length. In order to represent a straight line, graphs are most often used. And the graphs can be both in two-dimensional, and in three-dimensional system of coordinates. And they are built according to the coordinates of the points belonging to them. After all, if you look at a straight line, you can see that it consists of an infinite set of points.

However, there is something that the line is very different from other kinds of lines. This is her equation. In general, it is very simple, unlike, say, the equation of a circle. Surely, each of us passed it at school. But still write its general form: y = kx + b. In the next section, we will discuss in detail what each of these letters means and how to solve this simple equation of a straight line passing through two points.

The equation of the line

That equality, which was presented above, is the necessary equation for the straight line. It is worth explaining what here means that. As you can guess, y and x are the coordinates of each point belonging to a straight line. In general, this equation exists only because to each point of any line it is peculiar to be in connection with other points, and therefore there is a law linking one coordinate to another. This law determines how the equation of a straight line looks through two given points.

Why two points? All this is because the minimum number of points necessary to construct a straight line in two-dimensional space is two. If we take a three-dimensional space, then the number of points necessary for constructing a single straight line will also be equal to two, since three points already constitute a plane.

There is also a theorem proving that it is possible to draw a single straight line through two arbitrary points. This fact can be verified in practice by combining two random points on the graph with a ruler.

Now consider a concrete example and show how to solve this notorious equation of a straight line passing through two given points.

Example

Let us consider two points through which to construct a straight line. We give them coordinates, for example, M ₁ (2; 1) and M ₂ (3; 2). As we know from the school course, the first coordinate is the value along the OX axis, and the second is along the OY axis. Above, the equation of a straight line was introduced through two points, and in order for us to know the missing parameters k and b, we need to compile a system of two equations. In fact, it will be composed of two equations, each of which will have two of our unknown constants:

1 = 2k + b

2 = 3k + b

Now the most important thing remains: to solve this system. This is done quite simply. First, we express from the first equation b: b = 1-2k. Now we need to substitute the resulting equation into the second equation. This is done by replacing b by the equality obtained by us:

2 = 3k + 1-2k

1 = k;

Now that we know what the value of the coefficient k is equal to, it's time to find out the value of the next constant - b. This is made even easier. Since we know the dependence of b on k, we can substitute the value of the latter into the first equation and find out the unknown value:

B = 1-2 * 1 = -1.

Knowing both coefficients, we can now substitute them in the initial general equation of a straight line through two points. Thus, for our example we get the following equation: y = x-1. This is the desired equality that we should have obtained.

Before proceeding to the conclusion, let us discuss the application of this section of mathematics in everyday life.

Application

As such, the equation does not find a straight line through two points. But this does not mean that we do not need it. In physics and mathematics, the equations of lines and properties, which follow from them, are very actively used. You may not even notice it, but mathematics surrounds us. And even such seemingly unremarkable topics as the equation of a straight line through two points are very useful and very often applied at a fundamental level. If at first glance it seems that this can not come anywhere at all, then you are mistaken. Mathematics develops logical thinking, which will never be superfluous.

Conclusion

Now that we have figured out how to build lines at two given points, we do not need to answer any question related to this. For example, if the teacher says to you: " Write the equation of a straight line passing through two points," then you will not be able to do this. We hope that this article was useful to you.