Math Insight

Lines (and other items in Analytic Geometry)


Let's review some basic analytic geometry: this is description of geometric objects by numbers and by algebra.

The first thing is that we have to pick a special point, the origin, from which we'll measure everything else. Then, implicitly, we need to choose a unit of measure for distances, but this is indeed usually only implicit, so we don't worry about it.

The second step is that points are described by ordered pairs of numbers: the first of the two numbers tells how far to the right horizontally the point is from the origin (and negative means go left instead of right), and the second of the two numbers tells how far up from the origin the point is (and negative means go down instead of up). The first number is the horizontal coordinate and the second is the vertical coordinate. The old-fashioned names abscissa and ordinate also are used sometimes.

Often the horizontal coordinate is called the x-coordinate, and often the vertical coordinate is called the y-coordinate, but the letters $x,y$ can be used for many other purposes as well, so don't rely on this labelling!

The next idea is that an equation can describe a curve. It is important to be a little careful with use of language here: for example, a correct assertion is

The set of points $(x,y)$ so that $x^2+y^2=1$ is a circle.

It is not strictly correct to say that $x^2+y^2=1$ is a circle, mostly because an equation is not a circle, even though it may describe a circle. And conceivably the $x,y$ might be being used for something other than horizontal and vertical coordinates. Still, very often the language is shortened so that the phrase ‘The set of points $(x,y)$ so that’ is omitted. Just be careful.

The simplest curves are lines. The main things to remember are:

  • Slope of a line is rise over run, meaning vertical change divided by horizontal change (moving from left to right in the usual coordinate system).
  • The equation of a line passing through a point $(x_o,y_o)$ and having slope $m$ can be written (in so-called point-slope form) $$y=m(x-x_o)+y_o\;\;\;\;\;\hbox{or}\;\;\;\;\;y-y_o=m(x-x_o)$$
  • The equation of the line passing through two points $(x_1,y_1), (x_2,y_2)$ can be written (in so-called two-point form) as $$y={y_1-y_2\over x_1-x_2}(x-x_1)+y_1$$
  • ...unless $x_1=x_2$, in which case the two points are aligned vertically, and the line can't be written that way. Instead, the description of a vertical line through a point with horizontal coordinate $x_1$ is just $$x=x_1$$

Of course, the two-point form can be derived from the point-slope form, since the slope $m$ of a line through two points $(x_1,y_1), (x_2,y_2)$ is that possibly irritating expression which occurs above: $$m={y_1-y_2\over x_1-x_2}$$

And now is maybe a good time to point out that there is nothing sacred about the horizontal coordinate being called ‘$x$’ and the vertical coordinate ‘$y$’. Very often these do happen to be the names, but it can be otherwise, so just pay attention.


  1. Write the equation for the line passing through the two points $(1,2)$ and $(3,8)$.
  2. Write the equation for the line passing through the two points $(-1,2)$ and $(3,8)$.
  3. Write the equation for the line passing through the point $(1,2)$ with slope $3$.
  4. Write the equation for the line passing through the point $(11,-5)$ with slope $-1$.