Basic Equations for Lines

While there are many ways to write the equation for any given line on a cartesian coordinate system, there are really only a few which the average student of algebra really needs to remember in order to find success in more advanced mathematics. These are: Slope-Intercept Form, Point-Slope Form, and Intercept Form.

Slope-Intercept Form

Simply put, the equation most commonly used to denote the slope-intercept form of a linear equation is as follows:

y=mx+b.

This is surely the most common, and usually preferable form of linear equation, and more often than not, it is this equation that one will be attempting to find when analyzing a given line.

So what do these variables all represent?

The two most elements of this equation, as the name would suggest, are the variables which denote the slope and the line's y-intercept. In this equation, these variables are m and b, respectively. So, for a given equation, y=4x-8, the slope of the line in question is 4 and it crosses over the y-axis at -8.

(Slope, just as a reminder, is defind as rise/run; in other words, it is the ratio of the change in the value of y over change in the value of x. In this example, then, it might be more helpful to write 4/1 instead of 4, which implies that for every 1 unit the line moves to the right, it moves 4 units up. So this line is a rather steep line moving "uphill" from left to right. If the slope was a negative number, such as -4, then it would be moving "downhill")

Point-Slope Form

What if the problem doesn't provide the y-intercept, though? What if all that is available is the coordinates of a single point and the slope?

Simply use the point-slope equation:

(y-y1)=m(x-x1)

While this equation might seem more confusing than the first, it is really quite simple. Given a point on a line, such as (3,4), and the line's slope, perhaps it is 4 once again, simply plug these new coordinates in to replace x1 and y1:

(y-4)=4(x-3)

Then, using simple algebra, this equation can easily be transformed into the slope-intercept form:

y=4x-8

So, in essence, this equation described the very same line as the one above!

Now, as an additional note, point-slope form can also be used if the slope is not provided, but an extra point is. Given just two points, one can easily derive the slope by using the simple equation: (y-y1)/(x-x1), or, change in y over change in x.

Intercept Form

While most problems can generally be accomplished using some form of those first two equations, a third helpful one is the intercept form, which looks like this:

x/c + y/b =1

If an equation can be forced into this format algebraically, this form is actually quite helpful. Consider the following example:

x/2 + y/3 = 1

This equation gives us two very convenient points on the line in question: The x-intercept and the y-intercept, which are simply 2 and 3, respectively. So the line passes through points (2,0) and (0,3). From here it should be relatively simple to derive either of the previous two equations.

Now, with just these three rather simple equations put to memory, one has a wonderful foundation of knowledge to be used in solving the vast majority of linear equations out there.

Two Linear Equations

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