Hey everyone, it's Justin again. So far we've only learned how to write equations in slope-intercept
form. In this video, you'll learn how to use a new form, point-slope form.
Point-slope form is similar to slope-intercept form with a few major differences. The first
difference, as you can see, is that it's set up differently with a few different variables.
It still does involve the slope denoted by m, but instead of a y-intercept, we now have a
y1 and an x1. We've seen these variables before, and their meaning hasn't changed. Those are just
the x and y from some point on the line. That right there is the reason point-slope form is
so useful. With slope-intercept form, it has to be written using a specific point, the y-intercept,
and if it hasn't been given to you, you'll have to go through the trouble of solving for it.
But with point-slope form on the other hand, you can write it using any point on the line.
That even includes the y-intercept point. So any equation you could write in slope-intercept form
will be even easier to write in point-slope. Don't take my word for it though. Let's do
an example to see the difference. In order to write this equation in slope-intercept form,
first we have to plug in the slope. Then we plug in the point 2 for x and 3 for y. Then we solve
for b, and finally we get the equation that y =2x - 1. To write the equation in point-slope form,
we first plug in the slope. Then we plug in the point (2, 3) for x1 and y1, and that's it. The
equation in point-slope form is y - 3 = 2(x -2 ). We didn't have to do any solving or calculations
like we did for slope-intercept form, because any point is good enough for point-slope form.
What may not be immediately obvious is that both of those previous equations
were actually the same. To prove it, let's solve that point-slope version for y so that
it matches the slope-intercept form. First we'll distribute the slope and then add 3 to both sides,
and just like that the equation is in slope-intercept form. But since all we did
was manipulate the point-slope form, they're the same equation. Two forms of writing one equation.
One of the best times to use point-slope form is in cases like this, where the y-intercept on
a graph is kind of difficult to locate. We could tell it's somewhere between 0 and 1, but we don't
know exactly what the value is. So instead we choose two points that we can identify, and we
write the equation in point-slope form instead. The slope here is 1 over 3. So we could plug that
in. Now we just need to choose which point to use, and now remember, any point on the line works. If
we use this point, then the equation will be y - 2 = 1/3(x - 4). And if we use this point instead,
then the equation will be y - 3 = 1/3(x - 7). Both of these equations represent the exact
same line. The only difference is which point was used to write the equation. Try solving
both equations for y. You'll see they come up to the same equation in slope-intercept form.
Even in problems where all you've been given is two points, using point-slope
form is easy. Use the slope equation to find the slope, which in this case is -3/2,
then choose either of the points to use. If we choose the first point, we get this equation.
Or if we choose the second point, we get this equation. Again, remember that these are actually
the same equation for the same line. They're just expressed using different points on that line. In
fact, let's prove that these are the same equation by solving both for y. To make the calculations
easier, we'll put the slope in decimal form. So -3 divided by 2 is -1.5. For the version on the left,
we distribute the -1.5 and then we subtract 2 from both sides. In slope-intercept form,
this equation is the same as y = -1.5x - 0.5. For the version on the right, we distribute that slope
and then we add 4 to both sides. And would you look at that? They really are the same equation.
We've done a lot in this video with the new point-slope formula. We learned what it looks
like, compared it to slope-intercept form, and we even found equations in point-slope
form from both a graph and from a pair of points. But it all comes down to knowing
what you need for point-slope form, any point and a slope. In the next video,
we'll learn how the point-slope form can be used to solve word problems.