Understanding Point Slope Form

When you are graphing straight lines, one of the most common formats for describing the equation of the line is called "point slope form."  In this representation, the equation identifies one ordered pair that is on the line, and the slope.  If you were given only those pieces of information, you would have all that you would need to construct the line.  Continue reading to learn more about this line graphing concept, and make sure that you Like my post if it is helpful to you!

By using this formula...
- if you know one point that is on the line (x₁, y₁),
- and you know the slope of the line (m),
... with a little bit of mathematics and algebraic rearrangement, you can determine any other point (x, y) on the line.

Here's one type of problem that you will likely encounter:  Express in point slope form the line that passes through (4, 2) and has a slope of 8.  

To correctly solve this problem, all that you need to do is substitute the given values into the equation shown above.  Therefore, the correct expression is y - 2 = 8(x - 4).  It's as easy as that!

A more complicated problem would be something like this: What is the y-coordinate when x = 3 on the line that passes through (1, 1) and has a slope of 5?

To successfully work through this problem, you first approach it as you did the previous one.  That is, find the equation of the line.  In this case it is y - 1 = 5(x - 1).  Now, to find y when x = 3 is as simple as subbing in x = 3 into this equation, doing a bit of rearranging, and simplifying to isolate y.  Try it out for yourself, and you will see that y = 11 when x = 3.  In other words, the point (3, 11) is on the line that is described in the question.

Now that you've seen a few questions that can be asked about point slope form, perhaps it might help you to better understand the concept if you see what it actually means.  First off, consider all of the variables that are included in the expression:

We have an ordered pair (x₁, y₁), an unknown point (x, y) that can be any point on the line at all, and the slope (m).  Now, my question to you is: where have you seen these variables together in one place before?

If you answered that this is a rearrangement of the slope formula, then you get a gold star!  As we've seen before, the slope of a line is equal to rise over run.  In other words, the slope corresponds to the ratio of the change in vertical height to the change in horizontal distance of the line.  Mathematically, here is what you this means:

So, really, all we're really dealing with for any of this is the definition of the slope!  The point slope formula is just a different way of looking at it.

Now, it should be said that this may not be the most intuitive way of representing an equation of a line. Many people find expressing their equation in the form of y = mx + b to be more familiar and descriptive, since by definition it denotes the slope and y-intercept.  The intercept is a very easy "starting point" from which to extend your line, and with the known slope, is it simple to count spaces to plot another point.  Whichever way you express the equation of your line, assuming that the math is correct, they are just different ways of describing the same thing and so they aren't wrong... of course, unless your teacher specifically asks for one form or the other.

If you think about it, you should be able to see the connection between "point slope" and "slope intercept" forms.  Consider that in y = mx + b, the b term is the y-intercept (or y₁), which means that x₁ = 0.  So, you can say that:

y = mx + y₁

y - y₁ = mx

Looks familiar, except it is missing the x₁ term... but we already designated that it is 0 anyways!

Another way to show that these two formulas are the same thing is to equate them to the same variable, which therefore means the equations are equal.  This is like solving a system of equations.

If we solve for m in the point slope equation, we have: m = (y - y₁) / (x - x₁).
If we solve for m in the slope intercept equation, we have: m = (y - b) / x.

Since we're talking about the same line, it obviously is the same slope in each version, so it is fair to equate them. So:

(y - y₁) / (x - x₁) = (y - b) / x

Written like this, it is easy to see how the two equations correspond to each other.

A third way of representing a line's equation is to express it in standard form.  What this does is put everything on to one side of the expression, simplifies it, and sets it equal to 0.  Your expression will then have the form Ax + By + C = 0, where A is the "x" coefficient, B is the "y" coefficient, and C is the constant not associated with either x or y.

The whole point of all of this is to make a very simple point: all you need to fully describe the equation of a straight line is its slope and a point on it.  If you only have two points given, it is easy to calculate the slope from the slope formula, and then it is only a matter of plugging numbers into whichever expression you like.  If you only have the slope and no points, then you have the shape of a line but no point to which you can anchor it.  Get the slope, get a point, pick a way of expressing the equation of your line, and that's all there is to it.  If you made it all the way to the end, please remember to click the Like or +1 buttons (or both) below if you enjoyed this post!  Thanks!

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