Introduction


Welcome to the first lesson in the Straight Line series!

Over the next few (actually many) lessons, I’ll be covering the various forms of equations to a straight line, and a lot of related concepts, formulas and applications.

Without further ado, let’s get started !

To find the equation to a straight line, we’ll take a general point P(x, y) on the line, and find the relation between the coordinates, which will always hold true. (This in fact is the definition of an equation to any curve, as explained here).

We’ll start with the simple ones. Lines which are parallel to the axes.

1. Equation of a line parallel to the X axis

A line which is parallel to the X axis will always remain at a fixed distance from it. That is, the y-coordinate (i.e. the distance from the X axis) of any point P(x, y) will always remain the same (and will be equal to that distance)

Therefore the equation of a line parallel to the X axis will be of the form y = d, where d is the (signed) distance of the line from the X axis.

A line parallel to the X axis

2. Equation of a line parallel to the Y axis

Same here. The x-coordinate of a point on the line (i.e. its distance from the Y axis) will always remain the same. Therefore the equation be of the form x = d, where d is the (signed) distance of the line from the Y axis.

Lines parallel to the Y axis

3. Equation of the X axis and the Y axis

Well, what about the equations of the axes themselves? They are lines too, aren’t they? Of course they are !

Following the same method as above, the distance of the X axis from the X axis is… zero !

Therefore its equation is y = 0 .Similarly the equation of the Y axis is x = 0.

Equations of the coordinate axes

That’s it for now.

Note that, to uniquely describe a line, two conditions were required – the line being parallel to one of the axes, and being at a certain distance from that axis. I’ll come back to this in the subsequent lessons.

Lesson Summary

  1. The equation of a line parallel to the X axis will be of the form y = d, where d is the signed distance of the line from the X axis.
  2. The equation of a line parallel to the Y axis will be of the form x = c, where c is the signed distance of the line from the Y axis.
  3. The equation of the X axis is y = 0
  4. The equation of the Y axis is x = 0

The next lesson will deal with the equations of lines which are not parallel to any of the axes.


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