The angular coefficient in the equation is straight. The geometric meaning of the coefficient.

The angular coefficient in an equation of a straight line (also known as the slope or gradient) has a geometric meaning related to the inclination or steepness of the line. In simple terms, it determines how the line tilts or slants with respect to the horizontal axis on a graph.

The geometric meaning of the angular coefficient can be described as follows:

1. Positive Value: If the angular coefficient is positive, it means the line is slanting upward from left to right. In other words, as you move from left to right along the line, the y-values increase.

2. Negative Value: If the angular coefficient is negative, it means the line is slanting downward from left to right. In this case, as you move from left to right along the line, the y-values decrease.

3. Zero Value: When the angular coefficient is zero, the line is horizontal, and it does not tilt in either direction. The line is parallel to the x-axis.

4. Undefined or Infinite Value: If the angular coefficient is undefined or infinite, the line is vertical, and it does not tilt horizontally. The line is parallel to the y-axis.

So, the angular coefficient essentially quantifies the rate at which the y-coordinate changes concerning the x-coordinate as you move along the line. It provides information about the steepness and direction of the line in the Cartesian plane.

We transform the equation of the straight line ax + by + c = 0 to the form



We introduce the notation



Then we get y = kx + l.



Take two points on the line A (x1; y1) and B (x2; y2) such that x1 Their coordinates satisfy the equation of the line:



Subtracting these equalities term by term, we get



By drawing a straight line through point A parallel to the x axis and a straight line through point B parallel to the y axis, we get a triangle ABC. Notice that



If the line is located as follows:



Thats



Thus, the coefficient k in the equation of a straight line, up to a sign, is equal to the tangent of the acute angle that the straight line forms with the x axis.
The coefficient k in the equation of a line is called the slope of the line.