THE STRAIGHT LINE (ANALYTICAL GEOMETRY)

A straight line, the same as any curve contained completely in a plane is represented, as regards a system of Cartesian axes, by a function of two variables, any time the above mentioned function is capable of expressing the common condition that there satisfy absolutely each and everyone of the points that constitute the above mentioned line. For example, if we think about a straight line parallel to the axis of the abscissas, we need to begin for knowing where parallel happiness is planned, his equation counterfoil is and =mx b. With the application of Geogebra, we chart equation of the straight line to see his point of court better, as in this case in b, which is the independent term, in that m is a constant.

Function of the charted straight line is the following one y=3x 1, where m = 3 and b = 1 as we can see court in the axis and in 1. (Prepared for @dark69)

Equation of the straight line that happens for the origin of the system of coordinated Cartesian.

We are going now to demonstrate that quite straight that it happens for the origin of the system of coordinated it is represented by a function of the form y=mx or a function of two variables of the first grade, without independent term, in that m is a constant which meaning we will establish later. For this, we need to make to see that this function establishes or expresses the common condition to which there fit absolutely all the points that constitute a straight line that happens for the origin, in other words we must point out that the tidy one and of any point of the straight line really m is equal to the product of constant for the abscissa x of the above mentioned point, that is to say y=mx.


Using the application of Geogebra, we chart our equation y=mx.

The analysis of our equation we have following friends lover of the mathematics, We will begin for doing x=0 in the function, turning out to be like that y=0; this way there has a point O (0,0) that coincides with the origin of the coordinated ones. At once we give to the variable x another value, for example c, proving y=mc. (Prepared for @dark69)

Dependent on straight line (m).

We tackle it of the following way, with the intention of seeing the meaning of constant m we will allude to the straight line y=mx, which we will suppose that it forms an angle A positively, with regard to the positive sense of the axis of x.

On the straight line we take an any point
P (x, y), from which we plan the perpendicular one to the axis of x, y join the point of the origin with the any point P, to form the triangle rectangle, obtaining the following trigonometrical function: x and so A =; but of the proper given function and = m x, it is deduced that x and m = Substituting in the previous equality, is had: so A = m.


Constant m is the tangent trigonometrical one of the angle of inclination of the straight line that precisely receives the name of earring of the straight line, since it controls the biggest or minor inclination with regard to the axis of x. Taking in account that hanging m depends on an angle and that it is a coefficient of x in the function y=mx, also it can be called an angular coefficient of the straight line.

When constant m is positive, it indicates that the angle A of inclination of the straight line is sharp and, when it is negative, that the above mentioned angle measures more than 90 °, but without coming to 180 ° to exceed this value.


It is important to know that, The Analytical Geometry is convenient in calling parameters of a line, straight line or curve, to the constants that intervene in the representative corresponding function and of whose numerical values there depends the position that has happiness line, this independently of the proper name and ANALYTICAL GEOMETRY, the meaning of every constant. Consistently, the parameters of the straight line are hanging m and arranged the origin b, because they are these two constants on which the exact position of the straight line depends.

We know perfectly that the expression y=mx b is a function of two variables, but one tolerates to call it an equation of the straight line, because from the graphic point of view his solution is only a straight line.

Prepared for @dark69.

In conclusion we have, that this equation is used as a mathematical tool, in the field of others science as the Physics, for the study of of the movement, in the Astrology, to plan they distance in point in a straight line between the stars, in the technology, this tool is applied in the Gps to mark distances, and in the radars applied in the system of navigation.

Secondhand bibliography.
Analytical geometry and Trigonometry - Page 248 for Elena de Oteyza de Oteyza, ‎Emma Lam Osnaya, ‎Carlos Hernández Garciadiego – 2001.
Differential calculus And Integral - Page 21 – 2007.
Analytical geometry - Page 80 for René Jiménez – 2006.