How to use point-slope form when you are given two sets of points?

Question

Using point-slope form, find the equation of the following points: (2,3) and (5,1)

Answer 1

The point slope form is written like this:

y-y₁=m(x-x₁) where m is the slope, and (x₁,y₁) can be either one of the two points you are given.  First, we need to find the slope m from the given points.

The slope m=rise/run=(y₂-y₁)/(x₂-x₁). Let (2,3)=(x₁,y₁) and (5,1)=(x₂,y₂).  Then the slope is:

m=(1-3)/(5-2)=-2/3

Now put the point (2,3) in the point-slope equation along with the slope m to find the equation of the line between the two points.

y-3=(-2/3)(x-2)   Now distribute -2/3 through the parentheses, and add 3 to both sides

y=(-2/3)x+(4/3)+3

y=(-2/3)x+13/3    The term 13/3 is (4/3)+3 written in fraction form.  This is the equation of the line in a form called slope-intercept form.  In slope-intercept form, the slope is still -2/3, and the place where the line intercepts the y-axis is 13/3.

Hope that helps a bit!

Answer 2

Whitney, whenever you are wanting the equation for a straight line, you need to find two things:  How steep the line is (the slope) and its height when x=0 (the y-intercept).  In the end, there a few ways to represent this information, but a typical answer will look something like y = mx + b.

Point-slope form is one approach for finding a line and it requires two things also:  A point and a slope.  Pretty sweet name, huh?

1.  A point.  It looks like they gave you two to pick from here.  Pick either (2,3) or (5,1).  It doesn't matter which; the answer will come out the same.

2.  A slope.  Well, this question doesn't tell you the slope, so we'll need to calculate it.   Slope is calculated by the "rise-over-run" equation.  The rise is how much the y-coordinates changed, the run is how much the x-coordinates changed.  Looking at (2,3) and (5,1), how much did y change?  It went from 3 to 1, so it changed by -2 (it went down 2).  How much did x change? It went from 2 to 5, so it changed by 3 (it went up 3).  The slope is the change in y (rise) divided by the change in x (run).  The previous post has already done the arithmetic for you, but I'm sure you can do that on your own, too!

3.  Plug-n-chug.  Point-slope form looks like this:

y - y₁ = m(x - x₁)

The y₁ and x₁ are where you put the point that you picked in step one (remember that either point will be fine).  The m is the slope that you calculated in step 2.

At this point you are done! Awesome! You have it in point-slope form, which is what the question asked you to do.

Extra Credit Reading

4.  Slope-intercept form

But, could we do better? Probably.  Read the assignment heading carefully and see whether it asks you to convert your answer to slope-intercept form.

If it does, then there is one more simple step.  Slope-Intercept form is just another way to express the details of a line (kind of like having more than one word that means "awesome"). You need two things to use this form (you guessed it! A slope and an intercept).  Right now, you have something that looks like this:

y - y₁ = m(x - x₁)

All you want to do is have it look like this instead:

y = mx + b

Do you see that they both have an m? That means that number will stay the same.  So will y and x for the same reason.  To get b, all you do is take your point and slope and use this equation:

b = y₁ - mx₁

Fill in the x₁ and y₁ with the point you picked in step 1 and m with the slope you calculated in step 2.  Do the arithmetic, and this will give you the value of b.

5. It really doesn't matter which point I pick?

Nope! Try both, and you'll see that they both have the same slope-intercept form.