What is a line? A long straight thing that you draw?
Yes! But more precisely, it is a set of points that satisfy some sort of equation.
Notice there are two parts to that definition:
So, your choice is to either figure out how to describe a set of infinitely many things, or you can define one single equation to accomplish the same goal. I know what I’d rather do!
There are three forms of linear equations that you should be comfortable with:
The standard form of a line is the plain Jane, boring, but gets the job done form. It is written as Ax+By = C, where x and y are the variables and A, B, and C are constants and coefficients.
Note that this means that x and y can be anything you want, while A, B, and C are simply numbers that we don’t know. C is always a constant, since it is not attached to a variable, while A and B are both coefficients because they are attached to one of our variables.
Going back to the standard form of a line, we’d like to know what our constants and coefficients tell us. For example, if we have the line 3x+4y = 5, what do these numbers mean?
Unfortunately, there’s nothing immediately obvious that these numbers show us. That’s why I warned you that this form gets the job done and not much else!
However, if I told you that x=13, then you could substitute that into the equation for x to find that:
This shows us that y=1, but doesn’t help much beyond that.
The best (and most useful) form of a line is the slope-intercept form.
It looks like y = mx+b and the reason that this form is so useful is that it tells us two major features of our line:
The slope (how steep our line is)
The y-intercept (where it crosses the y-axis)
The best part about this is actually that as long as you remember the name of this form, you get the information in that order! This means that x and y are variables just like last time, but now we also have m as the slope of the line and b as the y-intercept.
For example, if I have the line y = 2x+6, I would immediately know that my slope is 2 and I cross the y-axis at y = 6. This form of the line is simple, straightforward and incredibly useful.
If that’s not the most useful information, then I don’t know what is.
The point-slope form of a line is the weirdest, but more immediately useful than standard form. Point-slope form of a line looks like y-y1 = m(x-x1).
The confusing part about this is the fact that we use x and y twice. Not to worry because we have a little 1 at the bottom right of two of them. This means that while x and y represent a variable that can be whatever you want, x1 and y1 represent the x and y-coordinate of one point in particular. Once again, m represents the slope in this equation too, thank goodness for mathematical consistency.
All of that might have sounded like gibberish, but it simply is a way of writing out the equation of a line using a single point and the slope!
For example, if I have a line with a slope of -3 and I know that the point (2, 3) is on my line, then I can just write out the equation immediately. Note that (x1,y1) = (2,3) and we can just substitute the values for slope and each coordinate of our point.
This gives a linear equation of y - 3 = -3 (x - 2). This equation may look weird, but it is useful!
As a final note, be aware that you need to be able to fluently translate between all three forms of a line because you will often be required to do so and demonstrate that you understand how all three of these are connected. Try writing the equation of a single line in each of these forms to find some interesting relationships; you may be quite intrigued to discover many of the patterns!