Mathematics is not a boring science, as it sometimes seems. It has a lot of interesting, although sometimes incomprehensible to those who are not eager to understand it. Today we will talk about one of the most common and simple topics in mathematics, or rather that of its field, which is on the verge of algebra and geometry. Let's talk about lines and their equations. It would seem that this is a boring school topic that does not bode well for anything interesting and new. However, this is not so, and in this article we will try to prove our point of view. Before moving on to the most interesting part 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 all this was needed and why now the knowledge of the following formulas also does not hurt.
History
Even in antiquity, mathematicians were carried away by geometric constructions and various graphics. It is difficult today to say who first came up with the equation of a line through two points. But it can be assumed that this person was Euclid - an ancient Greek scholar and philosopher. It was he who, in his treatise "Beginnings," gave rise to the foundation of future Euclidean geometry. Now this section of mathematics is considered the basis of the geometric representation of the world and takes place at school. But it is worth saying that Euclidean geometry acts only at the macro level in our three-dimensional dimension. If we consider the cosmos, then it is not always possible to imagine with the help of it all the phenomena that occur there.
After Euclid, there were other scientists. And they perfected and conceptualized what he discovered and wrote. In the end, it turned out a stable area of geometry, in which everything still remains unshakable. And for thousands of years it has been proven that the equation of a line through two points is very easy and simple to compose. But before starting to explain how to do this, we will discuss a little theory.
Theory
A straight line is a segment that is infinite in both directions and can be divided into an infinite number of segments of any length. In order to represent the direct, most often use graphics. Moreover, the graphs can be in either a two-dimensional or a three-dimensional coordinate system. And they are built according to the coordinates of the points that belong to them. Indeed, if we consider the line, we can see that it consists of an infinite number of points.
However, there is something that the line is very different from other types of lines. This is her equation. In general terms, it is very simple, unlike, say, the equation of a circle. Surely, each of us passed it at school. But still, we write its general form: y = kx + b. In the next section, we will examine in detail what each of these letters means and how to solve this simple equation of a line passing through two points.
Straight line equation
The equality that was presented above is the equation of the line we need. It is worth explaining what is meant here. As you might guess, y and x are the coordinates of each point on the line. In general, this equation exists only because it is common for each point of any line to be in connection with other points, and therefore there is a law linking one coordinate with another. This law determines how the equation of a straight line looks through two given points.
Why exactly two points? All this is because the minimum number of points needed to construct a straight line in two-dimensional space is two. If we take three-dimensional space, then the number of points needed to build one single line will also be two, since three points already make up the plane.
There is also a theorem proving that it is possible to draw a single line through any two points. This fact can be verified in practice by connecting two random points on the graph with a ruler.
Now we will consider a concrete example and show how to solve this notorious equation of a line passing through two given points.
Example
Consider two points through which you need to build a straight line. We give them the 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. The equation of the line through two points was given above, and in order to find out the missing parameters k and b, we need to compose a system of two equations. In fact, it will be composed of two equations, in each of which there will be 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, let us express b from the first equation: b = 1-2k. Now we need to substitute the resulting equality in the second equation. This is done by replacing b with the equality we obtained:
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 in the first equation and find out the unknown value:
b = 1-2 * 1 = -1.
Knowing both coefficients, we can now substitute them into the original general equation of the line through two points. Thus, for our example, we obtain the following equation: y = x-1. This is the desired equality that we should have received.
Before proceeding to the conclusion, we will discuss the application of this branch of mathematics in everyday life.
Application
As such, the equation of the line through two points does not find application. But this does not mean that we do not need it. In physics and mathematics, the equations of straight lines and the properties arising 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 prove to be very useful and very often applied at a fundamental level. If at first glance it seems that this cannot be useful anywhere, then you are mistaken. Mathematics develops logical thinking, which will never be superfluous.
Conclusion
Now that we’ve figured out how to construct lines along two given points, we don’t need to answer any question related to this. For example, if a teacher tells you: “ Write the equation of a line passing through two points,” then it will not be difficult for you to do this. We hope this article has been helpful to you.