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### Course: Get ready for AP® Calculus > Unit 2

Lesson 3: Point-slope form# Point-slope & slope-intercept equations

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). We can rewrite an equation in point-slope form to be in slope-intercept form y=mx+b, to highlight the same line's slope and y-intercept.

## Want to join the conversation?

- Why did Sal use the point (4,9)? Does it matter if you use (6,1) instead?(26 votes)
- No, it doesn't matter. You can use either point to find the equation of the line. Once the equation is to changed to slope-intercept form, the equation would match what Sal came up with.(41 votes)

- At1:36, Sal says that slope is change over x/change over y. I have learned this over and over, and now I finally get to ask my question: Why is it not change over x/change over y?(13 votes)
- It is not the change in X over the change in Y because X is always the independent variable in the situation, and Y is always the dependent variable in the situation. So when you are finding slope, you are trying to find the rate of change of the independent variable. So say for example that I am looking for christmas tree ornaments. Every ornament I buy increases the amount of money I spend by $2. The dependent variable is the amount of money because it changes depending on the amount of ornaments. So you already know what the dependent variable is. That is why you always do change in Y over change in X. You are searching for X. Otherwise, you would be searching for Y, and you already know what it is. So now I will solve this problem. I bought 2 ornaments, so my points for the graph would be the following:

( 1, 2 ) , ( 2, 4 ).

So now you need to find the slope. Remember, it is change in Y over change in X because you need to find the independent variable for the slope. To find slope you would do the following:

4-2

----- = 2/1 = 2.

2-1

So now you know the slope. Now you must find the y-intercept.

2=2(1) + b

2=2 + b

0=b

So now you know that there is no y-intercept in this problem. So you would set up the slope equation like so:

y = 2x + 0 or just y = 2x.

I hope that this helped! I have never answered a question before, so I'm sorry if this answer seemed to go on forever.

Sincerely - Anna F. 8th grade.(38 votes)

- Sal keeps saying arbitrary, what does it mean?(8 votes)
- Arbitrary basically means 'random'.

Here is a dictionary entry:*https://www.merriam-webster.com/dictionary/arbitrary*(14 votes)

- why didn't he plug in x=6 and y=1 for the point slope?(9 votes)
- You can absolutely do that! In fact, you can plug any point on the line and it will be correct. If you convert it to slope intercept form, you're gonna get the same answer in all cases--try it out! :)(10 votes)

- What if you have three points?(5 votes)
- If they are on the same line, you can use any two of the three and you should get the same answer.(9 votes)

- When the two points are (4, 9) and (6, 1), then to obtain m, how do we know whether to

(9-1)/(4-6)

or

(1-9)/(6-4)

?(4 votes)- Both are equivalent! Simplify the fractions to see it!

Can you figure out why the slopes turn out to be the same as long as we subtract both coordinates from each other in the same order regardless of the order we choose?(7 votes)

- Sir, everything is cleared but I have been still thinking that what is the difference btw two types SLOPE INTERCEPT and point slope form, what is the actual difference suppose if we have given one point (3,8) and (8,3) find the equation, it is cleared that is Point means we have to use point solve formula, but I saw somewhere many of the people using both types where is not giving and not asking about y-intercept, kindly response me on my this confusion?(5 votes)
- Point slope form and slope intercept form are both ways of expressing the equation of a straight line. Point slope form emphasizes the slope and ANY point on the line. Slope intercept form just shows the slope and the y-intercept of a line.

Finding the equation of the line using the points (3,8) and (8,3) [SLOPE INTERCEPT]

1. Find the slope between these points which is -1.

Equation then becomes y=-x+b. Solve for b by using one of the points above (I'll use (3,8))

y=-x+b

8=-3+b

b=11

Equation is y=-x+11

Finding the equation of the line using (3,8) and (8,3) [POINT SLOPE]

1. Find the slope which is -1. Input any point (I'll use 3,8)) into the equation.

y-8=-1(x-3)

2. That's the equation of the line in point slope form but you could put it into slope intercept form by performing distributive property.

y-8=-1(x-3)

y-8=-x+3

y=-x+11

Equation is y=-x+11(4 votes)

- What if you have fractions in the problem as your points and you have one zero as a y value?(3 votes)
- All of the rules still apply, no matter what numbers you have. So just do it the same as you would if you had whole numbers. Hope this helps.(5 votes)

- What happens if you are given a point not on the line then how could you figure it out(4 votes)
- That won't be a part of the equation.(1 vote)

- What if i don,t use (4,2) and instead i use (6,5)(3 votes)
- That's ok. Once you have the slope, you can use either point along with the slope to create the equation of the line. Once the equation is converted to slope-intercept form, the equation would match the one in the video.(4 votes)

## Video transcript

- [Voiceover] Let's say
we have a linear equation and we know when X is equal to four that Y is equal to nine and we've plotted that
point here on our XY plane. Actually, I forgot to label
the x-axis right over there. Now let's say we also know, we also know that when X is equal to six Y is equal to one. And we've plotted that point there. And so this green line
represents all of the solutions to this linear equation. Now what I want to do in this video is I want to say, well can we
find that linear equation and can we express it
in both point-slope form and in slope-intercept form. And I encourage you, like always, pause the video and see if you can do it. So let's first think
about point-slope form. Point, point-slope form. And point-slope form is
very easy to generate if you know a point on the line, or if you know a point that satisfies, where the X and Y coordinates
satisfy the linear equation, and if you were to know
the slope of the line that represents the solution
set of that linear equation. Now for sure we actually
were given two points that are solutions, that represent solutions to the linear equation. To fully apply point-slope, or
to apply point-slope easily, we just have to figure out the slope. And what we could do is,
we could just evaluate well what's the slope between
the two points that we know? And we just have to remind
ourselves that slope, slope, is equal to, slope is equal to change
in y over change in x. Sometimes people say rise over run. And what's that going to be? Well, if we say that this
second point right over here, if we say this is kind of our, if we're starting at this
point and we go to that point, then our change in Y, going
from this point to that point is going to be, it's going to be equal to one minus, one minus nine. One minus nine. This point right over here
is the point six comma one. So we started at Y equals nine, we finish at Y equals one, our change in Y is going
to be one minus nine. We have a negative eight change in Y, which makes sense. We've gone down eight. So this is going to be equal to, this is going to be
equal to negative eight. That's our change in Y. And what's our change in X? Well we go from X equals
four to X equals six. So we end up at X equals six and we started at X equals four. We started at X equals four, so our change in X is six minus four, which is equal to two. Which is equal to two. And you could have even done it visually. To go from this point to
this point your change in Y, your change in Y is you went down eight. So your change in, let me write this. So your change in Y is
equal to negative eight. And what was your change in X? Again to get to this point? Well, your change in X is positive two. So your change in X is equal to two. And so what's your slope? Change in Y over change in X. Negative eight over two
is equal to negative four. So now that we have a, now that we know the slope and we know a point, we know a, we actually know two points on the line, we can express this in point-slope form. And so let's do that. And the way I like to
it is I always like to just take it straight from the
definition of what slope is. We know that the slope between
any two points on this line is going to be negative four. So if we take an arbitrary
Y that sits on this line and if we find the difference
between that Y and, let's focus on this point up here. So if we find the difference between that Y and this Y, and nine, and it's over the difference between some X on the line and this X, and four. This is going to be the slope
between any XY on this line and this point right over here. And the slope between
any two points on a line are going to have to be constant. So this is going to be equal
to the slope of the line. It's going to be equal to negative four. And we're not in point-slope form or classic point-slope form just yet. To do that, we just multiply
both sides times X minus four. So we get Y minus 9, we get Y minus nine is equal to our slope, negative four times X minus four. Time X minus four. And this right over here is our classic, this right over here is our
classic point-slope form. We have the point, sometimes they even put
parenthesis like this, but we could figure out the point from this point-slope form. The point that sits on
this line with things that make both sides of
this equation equal to zero. So it would be X equals
four, Y equals nine, which we have right up there, and then the slope is right over here, it's negative four. Now from this can we now express this linear equation in y-intercept form? And y-intercept form, just
as a bit of a reminder, it's Y is equal to MX plus B. Where this coefficient is our slope and this constant right over here allows us to figure out our y-intercept. And to get this in this
form we just have to simplify a little bit of this algebra. So you have Y minus nine. Y minus nine is equal to, well let's distribute this negative four. And I'll just switch some colors. Let's distribute this negative four. Negative four times X is negative four X. Negative four time
negative four is plus 16. And now, if we just want to isolate the Y on the left hand side, we can add nine to both sides. So let's do that. Let's add nine, let's add nine to both sides. Let's add nine to both sides. On the left-hand side
we're just left with Y. And on the right-hand side we're left with negative four X and then
16 plus nine is plus 25. And there you have it. We have the same linear
equation, but it's now represented in slope-intercept form. Once again, we see the
slope right over here and now we can figure out
what the y-intercept is. The y-intercept when X is equal to zero, Y is going to be equal to 25. My axis right here, I
haven't drawn it high enough, but if I made it even taller
and taller and you see this line is gonna intersect the y-axis when Y is equal to, Y is equal to 25. So there you go, we wrote
it in point-slope form, that is that right over there, and we wrote it in Y, sorry, we wrote it in slope, we wrote it in slope-intercept form. Point-slope and slope-intercept. Hopefully you enjoyed that.