Linear equations and functions
Slopeintercept form
None
More examples of constructing linear equations in slopeintercept form
Linear Equations in Slope Intercept Form
Discussion and questions for this video
 In this video I'm going to do a bunch of examples of finding
 the equations of lines in slopeintercept form.
 Just as a bit of a review, that means equations of lines
 in the form of y is equal to mx plus b where m is the slope
 and b is the yintercept.
 So let's just do a bunch of these problems. So here they
 tell us that a line has a slope of negative 5, so m is
 equal to negative 5.
 And it has a yintercept of 6.
 So b is equal to 6.
 So this is pretty straightforward.
 The equation of this line is y is equal to
 negative 5x plus 6.
 That wasn't too bad.
 Let's do this next one over here.
 The line has a slope of negative 1 and contains the
 point 4/5 comma 0.
 So they're telling us the slope, slope of negative 1.
 So we know that m is equal to negative 1, but we're not 100%
 sure about where the yintercept is just yet.
 So we know that this equation is going to be of the form y
 is equal to the slope negative 1x plus b, where b is the
 yintercept.
 Now, we can use this coordinate information, the
 fact that it contains this point, we can use that
 information to solve for b.
 The fact that the line contains this point means that
 the value x is equal to 4/5, y is equal to 0 must satisfy
 this equation.
 So let's substitute those in. y is equal to 0 when x is
 equal to 4/5.
 So 0 is equal to negative 1 times 4/5 plus b.
 I'll scroll down a little bit.
 So let's see, we get a 0 is equal to negative 4/5 plus b.
 We can add 4/5 to both sides of this equation.
 So we get add a 4/5 there.
 We could add a 4/5 to that side as well.
 The whole reason I did that is so that cancels out with that.
 You get b is equal to 4/5.
 So we now have the equation of the line.
 y is equal to negative 1 times x, which we write as negative
 x, plus b, which is 4/5, just like that.
 Now we have this one.
 The line contains the point 2 comma 6 and 5 comma 0.
 So they haven't given us the slope or the yintercept
 explicitly.
 But we could figure out both of them from these
 coordinates.
 So the first thing we can do is figure out the slope.
 So we know that the slope m is equal to change in y over
 change in x, which is equal to What is the change in y?
 Let's start with this one right here.
 So we do 6 minus 0.
 Let me do it this way.
 So that's a 6 I want to make it colorcoded minus 0.
 So 6 minus 0, that's our change in y.
 Our change in x is 2 minus 2 minus 5.
 The reason why I colorcoded it is I wanted to show you
 when I used this y term first, I used the 6 up here, that I
 have to use this x term first as well.
 So I wanted to show you, this is the coordinate 2 comma 6.
 This is the coordinate 5 comma 0.
 I couldn't have swapped the 2 and the 5 then.
 Then I would have gotten the negative of the answer.
 But what do we get here?
 This is equal to 6 minus 0 is 6.
 2 minus 5 is negative 3.
 So this becomes negative 6 over 3, which is the same
 thing as negative 2.
 So that's our slope.
 So, so far we know that the line must be, y is equal to
 the slope I'll do that in orange negative 2 times x
 plus our yintercept.
 Now we can do exactly what we did in the last problem.
 We can use one of these points to solve for b.
 We can use either one.
 Both of these are on the line, so both of these must satisfy
 this equation.
 I'll use the 5 comma 0 because it's always nice when
 you have a 0 there.
 The math is a little bit easier.
 So let's put the 5 comma 0 there.
 So y is equal to 0 when x is equal to 5.
 So y is equal to 0 when you have negative 2 times 5, when
 x is equal to 5 plus b.
 So you get 0 is equal to 10 plus b.
 If you add 10 to both sides of this equation, let's add 10 to
 both sides, these two cancel out.
 You get b is equal to 10 plus 0 or 10.
 So you get b is equal to 10.
 Now we know the equation for the line.
 The equation is y let me do it in a new color y is equal
 to negative 2x plus b plus 10.
 We are done.
 Let's do another one of these.
 All right, the line contains the points 3 comma 5 and
 negative 3 comma 0.
 Just like the last problem, we start by figuring out the
 slope, which we will call m.
 It's the same thing as the rise over the run, which is
 the same thing as the change in y over the change in x.
 If you were doing this for your homework, you wouldn't
 have to write all this.
 I just want to make sure that you understand that these are
 all the same things.
 Then what is our change in y over our change in x?
 This is equal to, let's start with the side first. It's just
 to show you I could pick either of these points.
 So let's say it's 0 minus 5 just like that.
 So I'm using this coordinate first. I'm kind of viewing it
 as the endpoint.
 Remember when I first learned this, I would always be
 tempted to do the x in the numerator.
 No, you use the y's in the numerator.
 So that's the second of the coordinates.
 That is going to be over negative 3 minus 3.
 This is the coordinate negative 3, 0.
 This is the coordinate 3, 5.
 We're subtracting that.
 So what are we going to get?
 This is going to be equal to I'll do it in a neutral
 color this is going to be equal to the numerator is
 negative 5 over negative 3 minus 3 is negative 6.
 So the negatives cancel out.
 You get 5/6.
 So we know that the equation is going to be of the form y
 is equal to 5/6 x plus b.
 Now we can substitute one of these coordinates in for b.
 So let's do.
 I always like to use the one that has the 0 in it.
 So y is a zero when x is negative 3 plus b.
 So all I did is I substituted negative 3 for x, 0 for y.
 I know I can do that because this is on the line.
 This must satisfy the equation of the line.
 Let's solve for b.
 So we get zero is equal to, well if we divide negative 3
 by 3, that becomes a 1.
 If you divide 6 by 3, that becomes a 2.
 So it becomes negative 5/2 plus b.
 We could add 5/2 to both sides of the equation,
 plus 5/2, plus 5/2.
 I like to change my notation just so you get
 familiar with both.
 So the equation becomes 5/2 is equal to that's a 0 is
 equal to b.
 b is 5/2.
 So the equation of our line is y is equal to 5/6 x plus b,
 which we just figured out is 5/2, plus 5/2.
 We are done.
 Let's do another one.
 We have a graph here.
 Let's figure out the equation of this graph.
 This is actually, on some level, a little bit easier.
 What's the slope?
 Slope is change in y over change it x.
 So let's see what happens.
 When we move in x, when our change in x is 1, so that is
 our change in x.
 So change in x is 1.
 I'm just deciding to change my x by 1, increment by 1.
 What is the change in y?
 It looks like y changes exactly by 4.
 It looks like my delta y, my change in y, is equal to 4
 when my delta x is equal to 1.
 So change in y over change in x, change in y is 4 when
 change in x is 1.
 So the slope is equal to 4.
 Now what's its yintercept?
 Well here we can just look at the graph.
 It looks like it intersects yaxis at y is equal to
 negative 6, or at the point 0, negative 6.
 So we know that b is equal to negative 6.
 So we know the equation of the line.
 The equation of the line is y is equal to the slope times x
 plus the yintercept.
 I should write that.
 So minus 6, that is plus negative 6 So that is the
 equation of our line.
 Let's do one more of these.
 So they tell us that f of 1.5 is negative 3, f of
 negative 1 is 2.
 What is that?
 Well, all this is just a fancy way of telling you that the
 point when x is 1.5, when you put 1.5 into the function, the
 function evaluates as negative 3.
 So this tells us that the coordinate 1.5, negative 3 is
 on the line.
 Then this tells us that the point when x is negative 1, f
 of x is equal to 2.
 This is just a fancy way of saying that both of these two
 points are on the line, nothing unusual.
 I think the point of this problem is to get you familiar
 with function notation, for you to not get intimidated if
 you see something like this.
 If you evaluate the function at 1.5, you get negative 3.
 So that's the coordinate if you imagine that y is
 equal to f of x.
 So this would be the ycoordinate.
 It would be equal to negative 3 when x is 1.5.
 Anyway, I've said it multiple times.
 Let's figure out the slope of this line.
 The slope which is change in y over change in x is equal to,
 let's start with 2 minus this guy, negative 3 these are
 the yvalues over, all of that over, negative
 1 minus this guy.
 Let me write it this way, negative 1 minus
 that guy, minus 1.5.
 I do the colors because I want to show you that the negative
 1 and the 2 are both coming from this, that's why I use
 both of them first. If I used these guys first, I would have
 to use both the x and the y first. If I use the 2 first, I
 have to use the negative 1 first. That's why I'm
 colorcoding it.
 So this is going to be equal to 2 minus negative 3.
 That's the same thing as 2 plus 3.
 So that is 5.
 Negative 1 minus 1.5 is negative 2.5.
 5 divided by 2.5 is equal to 2.
 So the slope of this line is negative 2.
 Actually I'll take a little aside to show you it doesn't
 matter what order I do this in.
 If I use this coordinate first, then I have to use that
 coordinate first. Let's do it the other way.
 If I did it as negative 3 minus 2 over 1.5 minus
 negative 1, this should be minus the 2 over 1.5 minus the
 negative 1.
 This should give me the same answer.
 This is equal to what?
 Negative 3 minus 2 is negative 5 over 1.5 minus negative 1.
 That's 1.5 plus 1.
 That's over 2.5.
 So once again, this is equal the negative 2.
 So I just wanted to show you, it doesn't matter which one
 you pick as the starting or the endpoint, as long as
 you're consistent.
 If this is the starting y, this is the starting x.
 If this is the finishing y, this has to be
 the finishing x.
 But anyway, we know that the slope is negative 2.
 So we know the equation is y is equal to negative 2x plus
 some yintercept.
 Let's use one of these coordinates.
 I'll use this one since it doesn't have a decimal in it.
 So we know that y is equal to 2.
 So y is equal to 2 when x is equal to negative 1.
 Of course you have your plus b.
 So 2 is equal to negative 2 times negative 1 is 2 plus b.
 If you subtract 2 from both sides of this equation, minus
 2, minus 2, you're subtracting it from both sides of this
 equation, you're going to get 0 on the lefthand side is
 equal to b.
 So b is 0.
 So the equation of our line is just y is
 equal to negative 2x.
 Actually if you wanted to write it in function notation,
 it would be that f of x is equal to negative 2x.
 I kind of just assumed that y is equal to f of x.
 But this is really the equation.
 They never mentioned y's here.
 So you could just write f of x is equal to 2x right here.
 Each of these coordinates are the coordinates
 of x and f of x.
 So you could even view the definition of slope as change
 in f of x over change in x.
 These are all equivalent ways of viewing the same thing.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

Have something that's not a question about this content? 
This discussion area is not meant for answering homework questions.
I don't understand why we divide by three at about 7:50. help?
He is basically simplifying the problem 5/6 multiplied by 3/1, 3 and 6 both are divisible by 3 so he simplifies the equation making it 5/2 multiplied by 1 which then results in 5/2
At 7:50 he is multiplying 5/6 by 3 which is the same as 3/1. So if you multiply the numerators you would get 15 and when you multiply the denominators you get 6. Hence you are left with 15/6 which is not simplified. Because the numerator and denominator are both divisible by three he divides them by three to simplify the fraction and he gets 5/2
I just learned this in math but you divide the number by 3 because as seen above it shows...
"05 over 33. This means that it equals 5/6=5/6. 5 and 5 would go to 1=1 and 6=6 would simplify to 3=1. SO that is what he means as divide by 3. :)
"05 over 33. This means that it equals 5/6=5/6. 5 and 5 would go to 1=1 and 6=6 would simplify to 3=1. SO that is what he means as divide by 3. :)
that makes no sense
how do you change from pointslope form to slopeintercept?
y  y1 = m(x  x1)
=> y  y1 = mx  mx1
=> y = mx  mx1 + y1
then simplify.
With an example containing actual numbers, you see it's much easier than the steps above imply:
y  7 = 4(x  2)
=> y  7 = 4x  8
=> y = 4x  8 + 7
=> y = 4x  1
Additional resource: http://www.mathwarehouse.com/algebra/linear_equation/pointslopetoslopeinterceptform.php
=> y  y1 = mx  mx1
=> y = mx  mx1 + y1
then simplify.
With an example containing actual numbers, you see it's much easier than the steps above imply:
y  7 = 4(x  2)
=> y  7 = 4x  8
=> y = 4x  8 + 7
=> y = 4x  1
Additional resource: http://www.mathwarehouse.com/algebra/linear_equation/pointslopetoslopeinterceptform.php
yy1=m(xx1).....this is the point slope form
yy1=x(m)+x1(m)....you distribute the m
y=xm+x1(m)+y1.....you move the y1 to the other side
and then simplify
ex:y+1=1/2(x+6)....this is the point slope form
y+1=1/2x3
y=1/2x3+2
y=1/2x1....that is the slope intercept form
yy1=x(m)+x1(m)....you distribute the m
y=xm+x1(m)+y1.....you move the y1 to the other side
and then simplify
ex:y+1=1/2(x+6)....this is the point slope form
y+1=1/2x3
y=1/2x3+2
y=1/2x1....that is the slope intercept form
the formula for pointslope form is yy1=m(xx1) if you had a slope of 3, and a point on the line that was (1,2) it would look like y2=3(x1) which would simplify to y2=3x3 you add 2 to both sides and it becomes: y=3x5
Why do we use X as the common variable in algebra? X is also the horizontal axis in graphs.
That's not a coincidence, as you will see when you start graphing algebraic equations.
The whole point of equations are to graph a line and show a relationship between two of more values so x is one of the values that you a showing a relationship to
I think x, y, and z are all common variables, but the first letter(x) seems to be used a lot in algebraic equations.
When you start graphing you will understand
Thanks a lot guys..
Wouldn't they be mistaken for each other?
What is the difference between f(x) and y? Sal said that he assumed they were equal, but can there be cases where they aren't? How would that work?
They're the same thing, but in higher math classes you use f(x) because you graph functions and y can be used as a variable. Say you have a question that says
f(x)=3x + 4, what is f(2), you would simply plug in 2 for x and get 10. It's much more useful in more advanced math.
You can also watch this video.
https://www.khanacademy.org/math/cceighthgrademath/cc8threlationshipsfunctions/cc8thfunctionnotation/v/differencebetweenequationsandfunctions
At first when you are learning functions, you would think that they are silly and you can use y instead. But later on, you will realize that y is silly.
I hope this helps!
f(x)=3x + 4, what is f(2), you would simply plug in 2 for x and get 10. It's much more useful in more advanced math.
You can also watch this video.
https://www.khanacademy.org/math/cceighthgrademath/cc8threlationshipsfunctions/cc8thfunctionnotation/v/differencebetweenequationsandfunctions
At first when you are learning functions, you would think that they are silly and you can use y instead. But later on, you will realize that y is silly.
I hope this helps!
at 3:27 shoudnt it be y2y1 over x2x1 instead of y1y2 over x1x2
Believe it or not, they are the same thing. Try it.
Why is 6/3 equal to 6/3 or 2?
6/3 means 6 ÷ 3
A positive divided by a negative is a negative, and 6 ÷ 3 = 2.
Therefore 6/3 = 2
6/3 means 6 ÷ 3
A negative divided by a positive is a negative, and 6 ÷ 3 = 2
Therefore 6/3 = 2
2 = 2
Since 6/3 = 2, 6/3=2, and 2=2 (i.e. they are all equal to the same thing), they must also be equal to each other.
This same process could be done for any numbers meaning that
(x)/y = x/(y) = (x/y)
or in words: A negative in a fraction can go in front, in the numerator, or in the denominator, but not both.
A positive divided by a negative is a negative, and 6 ÷ 3 = 2.
Therefore 6/3 = 2
6/3 means 6 ÷ 3
A negative divided by a positive is a negative, and 6 ÷ 3 = 2
Therefore 6/3 = 2
2 = 2
Since 6/3 = 2, 6/3=2, and 2=2 (i.e. they are all equal to the same thing), they must also be equal to each other.
This same process could be done for any numbers meaning that
(x)/y = x/(y) = (x/y)
or in words: A negative in a fraction can go in front, in the numerator, or in the denominator, but not both.
When you simplify it you'll get 2 on both of them
to find the slope sal subtracts (at 3:00) 60/25. check. next problem i paused, tried to figure it out myself and subtracted (at 5:54) 50/3(3) to find the slope. sal however reversed the order and did 05/33. we ended up with very different answers. how do i know which point to subtract from which point? he did them in two different orders.
Ramona, it looks like you are on the right track. Continuing from your set up, you should get this result:
slope = (50)/(3(3)) = 5/(3+3) = 5/6
That's the same answer Sal got to in the video.
To answer your question about knowing which point to subtract from which point  either way is correct! It is _your choice_ which is the 'first' point and which is the 'second'  just make sure you make the _same_ choice for both the x and y coordinates. In the video, Sal chose to set up the calculation 'green minus orange' and you set it up 'orange minus green'. But either way, once all the arithmetic is done, you should have the same answer. (Why? Note what happens at 7:01: _"...the negatives cancel out."_
slope = (50)/(3(3)) = 5/(3+3) = 5/6
That's the same answer Sal got to in the video.
To answer your question about knowing which point to subtract from which point  either way is correct! It is _your choice_ which is the 'first' point and which is the 'second'  just make sure you make the _same_ choice for both the x and y coordinates. In the video, Sal chose to set up the calculation 'green minus orange' and you set it up 'orange minus green'. But either way, once all the arithmetic is done, you should have the same answer. (Why? Note what happens at 7:01: _"...the negatives cancel out."_
At 8:37, if b in the equation y = mx+b is 5/2 or 5 halves, how would it be plotted on the coordinate plane in fraction form as the y intercept? Isn't the y intercept supposed to be a whole number, so that it can be plotted on the coordinate plane?
The yintercept can be any number, it need not be a whole number (usually is not). You just plot 2.5 as best you can, halfway between 2 and 3.
This isn't exactly related to the topic, but why does 60/25 become 6/3 ?
Why isn't it just 6/3 ? I never understood this and it always stuffs me up.
Why isn't it just 6/3 ? I never understood this and it always stuffs me up.
Tomm0,
Your mistake is that (60) is +6, not 6
If you had a line that included the points (6,2) and (0,5), the slope would be (change in y)/(change in x) which would be (60)/25)
(60) is +6 , not 6
(25) is 3
So the fraction is 6/3
which reduces to 2/1 = 2
I hope that helps make it click for you.
Your mistake is that (60) is +6, not 6
If you had a line that included the points (6,2) and (0,5), the slope would be (change in y)/(change in x) which would be (60)/25)
(60) is +6 , not 6
(25) is 3
So the fraction is 6/3
which reduces to 2/1 = 2
I hope that helps make it click for you.
u need to respect the man he is smart and he is also human so are you so treat everyone the way u want to be treated
How Would You Work This Type Of Problem When It's Set Up In A Table???
do you mean if its like
x2/3/4/5/6
y4/6/8/10/12 you have to find a pattern (in this case y is twice the amount of x) so y=2x
x2/3/4/5/6
y4/6/8/10/12 you have to find a pattern (in this case y is twice the amount of x) so y=2x
Choose any two points, then find slope the same way you did without the table. After you find your slope, plug any one of the points back into y = mx + b and solve for b.
X  Y
1  3
2  5
3  7
Points (1,3),(2,5),(3,7)
Using the points (1,3) and (2,5)
m = variation in y / variation in x = 3  5 / 1  2 = 2 / 1 = 2
y = mx + b
y = 2x + b
Using the point (1,3)
(3) = 2 . (1) + b
Subtracting 2 on both sides:
b = 1
So the equation for this line is:
y = 2x + 1
1  3
2  5
3  7
Points (1,3),(2,5),(3,7)
Using the points (1,3) and (2,5)
m = variation in y / variation in x = 3  5 / 1  2 = 2 / 1 = 2
y = mx + b
y = 2x + b
Using the point (1,3)
(3) = 2 . (1) + b
Subtracting 2 on both sides:
b = 1
So the equation for this line is:
y = 2x + 1
there is always a slope #
I don't get it. At 3:39 is confusing. Why can't we use the slope equation to get the slope and then the slope would be 2 not 2. Am I missing something? He said if it's switch then it would be be negative?
I think I understand what you are trying to say, if you are talking about the equation y2y1/x2x1. So you would have 06/52. First, if we think about it, the slope will be the same no matter if it is from (2,6) to (5,0) or (5,0) to (2,6). Just know the slope is the same, so we can do it Sal's way of 60/25 or the way 06/52. You probably thought that 06 was just 6, but 06 is 6. Then you get 52 which is 3, so 6/3. If you thought 06 was 6, you would have got 2 as the slope. I think that is what you are asking
Comment if you get it or if you meant something else
Comment if you get it or if you meant something else
OMG that is what I meant! Thank you! I forgot that the number would be negative 6. I can't believe I didn't see it!
write th equation of the line in slope intercept form if the slope is 1/5 and the yintercept is 9
when you put an equation in y intercept form you always start with y= the slope goes with the x and the intercept is the next term.
y = mx +b
m is the slope, b is the intercept
y= 1/5x 9
y = mx +b
m is the slope, b is the intercept
y= 1/5x 9
What if the slope is undefined, but you are given a point. What do you do?
Hannah,
If you have a linear equation where the slope is undefined, then the denominator of the slope must be 0 since anything divided by 0 is undefined.
The denominator of the slope is the change in x
So the change in x is zero.
So if your point was (1,3) and the slope is undefined, you know that some other point has a change in y but x changes by zero,
So x would always by 1.
Points (1,3), (1,4), (1,5) would all be on your line.
The equation of this line is x=1+0y or x=1
If you graphed it you would have a vertical line. going through the x axis at the point given as the x value in your original point.
I hope that makes it click for you.
If you have a linear equation where the slope is undefined, then the denominator of the slope must be 0 since anything divided by 0 is undefined.
The denominator of the slope is the change in x
So the change in x is zero.
So if your point was (1,3) and the slope is undefined, you know that some other point has a change in y but x changes by zero,
So x would always by 1.
Points (1,3), (1,4), (1,5) would all be on your line.
The equation of this line is x=1+0y or x=1
If you graphed it you would have a vertical line. going through the x axis at the point given as the x value in your original point.
I hope that makes it click for you.
if you have 2lbs of jellybeans for $9, what of .5 lbs? 3lbs? 4 lbs?
I would solve this using proportions: if you have the proportion $9 per 2 lbs., you can divide both by 2 to get $4.50 for 1 lb. From here you can multiply both by 3, 4, or 1/2 to get your answers. I'll leave that up to you ;D
at 1:48 how did he find the yintercept of the equation if it was not written in standard form ?
He knew that line goes through the point `(4/5, 0)`, right? `4/5` is an `x` value and `0` is an `y` value of this point. Also he knew that slope (m) is equal to `1`. Therefore by substituting the values to the formula of `y = mx + b` we get `0 = 1 * 4/5 + b` which is an equation with one variable and just like Sal did  `0 = 4/5 + b` therefore `b = 4/5`.
what do i do when the equation is Y = X
it means that whatever your X value is,that will also be your Y value. when X=1 Y=1 when X=5 Y=5.and therefore the slope is 1(1/1). it follows the usual equation of a line Y=mX + b.when X is 0 so is Y, that's why the Y intercept (b) isn't in the equation, its 0.
AT 2:23 I don't get how he got x
Try watching the video again.
i agree watch the video again
How do I write the slopeintercept form of the equation of the line through: (3,2), if the slope is undefined?
Hi,
If a slope is undefined, it means that the x value does not change. Remember that slope is calculated as the change in the y value divided by the change in the x value. If x does not change, then we are dividing by zero, which is undefined. Therefore, an undefined slope is a vertical line through the point you were given.
You cannot really write it in a correct slope/intercept form. Usually this is written as:
x=whatever the x value of your point is.
In your case, the equation would be
x=3
And the graph would be a vertical line running through the 3 on the x axis.
Hope that helps :)
If a slope is undefined, it means that the x value does not change. Remember that slope is calculated as the change in the y value divided by the change in the x value. If x does not change, then we are dividing by zero, which is undefined. Therefore, an undefined slope is a vertical line through the point you were given.
You cannot really write it in a correct slope/intercept form. Usually this is written as:
x=whatever the x value of your point is.
In your case, the equation would be
x=3
And the graph would be a vertical line running through the 3 on the x axis.
Hope that helps :)
I didn't understand the equation f(1.5)=3,f(1)=2?
This is because our function here is
*f(`x`)=(`x`)2*
So each time we put in a value of `x` we multiply it by 2
Ex.
f(`1.5`)=(`1.5`)*2=3
f(`1`)=(`1`)*2=2
*f(`x`)=(`x`)2*
So each time we put in a value of `x` we multiply it by 2
Ex.
f(`1.5`)=(`1.5`)*2=3
f(`1`)=(`1`)*2=2
how do you find a y intercept in a graph
i don't understand what is going on when you have to solve for b, can i please have some help?
When solving for b we put the values (x,y) of the given point (which sit on the line) in the the equation of the line in its general form: y=mx+b, where m is our slope (in this example 1) Than we separate b on one side of the equation and numbers on another.
how is it " m=delta y over delta x? " why is it not delta x over delta y?
slope can also be though of in terms of rise over run. since delta y is how far the graph is rising and delta x is how far the graph "runs", thats why the slope is delta y over delta x. It is just a mathematical convention that slope is measured that way.
muchos gracias
Is it m=y2y1 over x2x1 or m=y1y2 over x1x2? I need help. Please and Thank You.
It really doesn't matter. As long as you have the y's over the x's, you'll get the slope of your line. In fact, the two ratios are equal anyways because y1y2=(y2y1), annd the same for the x's, so multiply one way by 1/1, and you'll get the other.
It doesn't matter! Both end up with the same thing  try it. An example: (35) / (27) = 2 / 5 = 2/5. Or (53) / (72) = 2/5.
Okay thank you
About the substitution of the points. What if you don't have a zero? Do you just go with the points that are closest to one?
@ 1:07} Do you have to use m to represent the slope or do you choose any variable?
Well technically, you have to use m to represent the slope or else it wouldn't be right.
How do you make it easier to convert from a problem such as 5x + 2y = 6 without getting confused over negative numbers?
Tommy,
Typically to isolate y (so you can get into the form y = mx + b) you would start this problem by adding 5x to both sides (to get rid of the 5x on the left). This first step would give you:
2y = 5x + 6
Bingo. Your negative sign is gone. Now a you divide both sides by 2:
y = 5/2x + 3 ... and you're done!
Typically to isolate y (so you can get into the form y = mx + b) you would start this problem by adding 5x to both sides (to get rid of the 5x on the left). This first step would give you:
2y = 5x + 6
Bingo. Your negative sign is gone. Now a you divide both sides by 2:
y = 5/2x + 3 ... and you're done!
at 3:53 , why does Sal put a negative when 6  0 is not negative?
because he went the other way around just to prove you would get the same value for the slope.
how do i find an equation of the slope line?
I do not get your question.
it is y1 y2 over x1 x2
it is y1 y2 over x1 x2
( y₁ y₂)/( x₁ x₂) is the formula for slope.
to find the equation of a line sub (x,y) of a line into y=mx+b form
solve to find b
and write down the equation of the line
to find the equation of a line sub (x,y) of a line into y=mx+b form
solve to find b
and write down the equation of the line
How do you solve a problem when it asks you, " Write the equation that describes each line in slopeintercept form." and the problem is [slope= 2/7, (14,3) is on the line]? PLEASE ANSWER SOON OR PUT A VIDEO UP ON KHAN ACADEMY ABOUT MY QUESTION!!!!!!!
well look at it this way: slope= 2/7, (14,3), look at in the video 10: 11 through the rest of the video, I hoped this helped you Sophia.
And when graphing these can if you can switch the 6/3 around will it change the graph points
Stephanie,
If I understand your question correctly, you are asking how to find slope. If you are given points, (example: 1,2, 3,4) then you can find your slope using the equation:
Y2  Y1 over X2X1.
In this case, Y2 is 4 because this is the second y in the two points, Y1 is 2, X2 is 3, and X1 is 1. We now have: 42 over 31 which is 2 over 2 which can be simplified to 1. The slope of the equation is 1.
Hope this helped!
If I understand your question correctly, you are asking how to find slope. If you are given points, (example: 1,2, 3,4) then you can find your slope using the equation:
Y2  Y1 over X2X1.
In this case, Y2 is 4 because this is the second y in the two points, Y1 is 2, X2 is 3, and X1 is 1. We now have: 42 over 31 which is 2 over 2 which can be simplified to 1. The slope of the equation is 1.
Hope this helped!
Isn't 21=3 ? Sal said that it equals to 2?!
The answer is not 2. Probably just a simple mistake
The answer is negative three.
2 + 1= 3
It's the same as 2 1= 3
2 + 1= 3
It's the same as 2 1= 3
Yeah sorry I meant 3 but he said 2.
what is the *slopeintercept formula*
and x and y are points on the graph.
y=mx+b
m is the slope and b is the yintercept.
m is the slope and b is the yintercept.
the formula is y=mx+b
At 6:39, i understand that Sal started with the second set of coordinates (3,0) first but, i thought it didn't matter. I later tried doing the equation with the first set of coordinates first (3,5) and it doesn't work because you end up dividing 5 by 0. This doesn't work. Can someone explain how to know which coordinate set to use first? I am a little confused. Thank you, Elena
You were initially right: the order doesn't matter when finding slope.
(5 0)/(33) = 5/6
(05)/(33) = 5/6 = 5/6
Both give the same answer.
(5 0)/(33) = 5/6
(05)/(33) = 5/6 = 5/6
Both give the same answer.
At 6:11, how come its 05 instead of 50? Is that crucial to get the correct answer or does it not matter, because you are still finding the change?
(y2  y1)/(x2x1) works either way. Just make sure that whatever point you started with for the difference in y in the numerator is the same point you start with for the difference of x in the denominator.
i really do not understand this video. how would you write in a equation form: Jan wants to buy maps and atlases. the maps cost $2 each and the atlases cost each $5. If she buys 3 atlases and spends $25, how many maps can she buy?
Can you plze explain how to write an equation with this example?
Can you plze explain how to write an equation with this example?
An equation for this is
t = 2m + 5a
where t = total, m = maps, a = atlases
You're also told that a = 3 and t = 25, so
25 = 2m + 5 * 3
Solve for m.
t = 2m + 5a
where t = total, m = maps, a = atlases
You're also told that a = 3 and t = 25, so
25 = 2m + 5 * 3
Solve for m.
Will you please show how to find the slopeintercept form for the line satisfying the following:
xintercept 3, yintercept 2/3?
xintercept 3, yintercept 2/3?
Well remember x represent the horizontal line and so your first point will be (3,0) while the vertical line (y) represent a point in y lower than 1 (dividing 1 in thirth so you take 2 of them and you have 2/3). Finally your second point is (0,2/3) and join the dots.
m= 3  0/0  2/3= 9/2 = 4 1/2... y= 9/2x+2/3 (mx+b)
Sorry if I was a little obvious but my intention was be clear!
m= 3  0/0  2/3= 9/2 = 4 1/2... y= 9/2x+2/3 (mx+b)
Sorry if I was a little obvious but my intention was be clear!
How do you write the the equation if its a word problem?
Write down what you know and what you don't know and what it is asking for clearly. Try to find a relationship. Remember that all lines have a slope (m), which is basically how steep it is, which is simply a measure of how far it moves along the x axis over how far it moves along the y axis, and a point of origin where it crosses a known point. In normal notation, this is the y intercept (b), which occurs when x = 0.
what is the x for ?
Its the value of the number on the x axis.
Oh yeah,  x is just a variable.
Oh yeah,  x is just a variable.
noni la cubana, in y = mx + b, x just represents a general variable! x could be anything. That's why you keep it as x when you write an equation in "Slope Intercept Form".
Hope that helps!
Hope that helps!
If we can plot an equation in which no x is defined ( e.g. 2y=8 where we assume 0x) why can we not graph an equation in which no y is defined (e.g. 4x=8)
You can graph 4x=8. When you divide both sides by 4 you get x=2. This is simply a vertical line which crosses the x intercept at 2. That vertical line includes all points in which x is equal to 2.
You can, it is the point 2 on the x axis. If plotted in two dimensions (2,0).
Is there a particular reason why Sal used less common variables in this video? Or did he just want to through in a variety?
Well, if you are talking about him using the variables: m, x, y, and b, he is using the equation of slopeintercept form, which is *y = xm + b*. "b" is the yintercept and "m" is the slope of the line, and x + y are points on the line, of course. This equation makes a straight line. If you want to know more, just look here: https://www.khanacademy.org/math/algebra/linearequationsandinequalitie/equationofaline/v/graphingalineinslopeinterceptform.
Also, the f at around 10:32 is a symbol for function.
Also, the f at around 10:32 is a symbol for function.
at 2:57 in the videos sal used these triangular shaped things in the equation what are they and what do they represent
They are the Greek symbol delta. They are accepted in math and science as the word "change".
thanks for telling me what they ment cause i started to get a bit skeptical about them.
How do you write an equation in slope intercept form?
A linear equation in slopeintercept form is y = mx + b. Where "m" is your slope and "b" is your yintercept.
How to graph y=x
The yintercept would be zero and the slope would be 1.It would have a constant increase and go in a straight equal line.
Hi you have done that in a problem
Line contain points (2,6) and (5,0)
m=y1y2/x1x2
but the formula is
http://cs.selu.edu/~rbyrd/math/slope/slo_eq1.gif
Line contain points (2,6) and (5,0)
m=y1y2/x1x2
but the formula is
http://cs.selu.edu/~rbyrd/math/slope/slo_eq1.gif
Since (x1,y1) and (x2,y2) could be either of the two points, it it true that is doesn't matter which way the slope formula is written. It is important that you use your original point for both the numerator and denominator or your slope will have the opposite sign it should have.
It doesn't matter if its y1y2 or y2y1
Around 4:12, Sal had a problem which said,
"The line contains points (2,6) and (5,0)"
In this problem, Sal subtracts this way, "60/25"
In the next problem 6:48 Sal does his problem "The line contains points (3,5) and (3,0)" and SWITCHES AROUND compared to his problem before, by subtracting "05/3  3".
I'm not a mathematical genius compared to Sal so please help me see where I'm going wrong. I thought the problem would turn out like this,
"50/ 33"
Sal's Equation: "05/ 3 3"
"The line contains points (2,6) and (5,0)"
In this problem, Sal subtracts this way, "60/25"
In the next problem 6:48 Sal does his problem "The line contains points (3,5) and (3,0)" and SWITCHES AROUND compared to his problem before, by subtracting "05/3  3".
I'm not a mathematical genius compared to Sal so please help me see where I'm going wrong. I thought the problem would turn out like this,
"50/ 33"
Sal's Equation: "05/ 3 3"
Christian, that's ok! You see, when you want to do these problems, you usually have two sets of points, right? Let's call them (X1,Y1) and (X2, Y2). Now, in school, usually the general formula taught for finding slope is (Y2  Y1) divided by (X2  X1). But, as you'll learn later on (if you aren't taking higher level math yet) that the equation is actually "CHANGE OVER Y" divided by "CHANGE OVER X". The order in which you subtract doesn't matter, as long as you subtract the Ys and divide them by the subtraction of the Xs. If you'll notice, your equation gives you the answer "5/6," which is "5/6". Sal's answer is also "5/6".
Hope that helps!
Hope that helps!
what does delta mean?
Delta is a Greek letter: http://en.wikipedia.org/wiki/Delta_(letter).
Mathematicians use the letter delta to represent a change in value of a variable. For example, ∆x is the change in the x variable. So when you see ∆y/∆x you know you're dealing with slope as it's the change in y compared to (or over) the change in x. This is also called rise over run: rise is how far you change vertically. Run is how far you change horizontally.
Mathematicians use the letter delta to represent a change in value of a variable. For example, ∆x is the change in the x variable. So when you see ∆y/∆x you know you're dealing with slope as it's the change in y compared to (or over) the change in x. This is also called rise over run: rise is how far you change vertically. Run is how far you change horizontally.
Why do these equations have the slope inside? Is it essential?
yes, a slope in a linear equation is mandatory because it can tell you about the rate of change of a line. The slope can make a line. Essentially, the slope is compulsory for linear equations and further understanding of slope can lead you into calculus.
what is linear equation
Yogi,
A linear equation is an equation which graphs to form a line.
It usually takes the form of y = ax+b where a and b are constants such as y=2x+1.
When you graph it, it forms a strait line, so they call it a linear equation.
Here is a video that might help you understand more:
http://www.khanacademy.org/math/algebra/linearequationsandinequalitie/graphing_solutions2/v/algebragraphinglines1
A linear equation is an equation which graphs to form a line.
It usually takes the form of y = ax+b where a and b are constants such as y=2x+1.
When you graph it, it forms a strait line, so they call it a linear equation.
Here is a video that might help you understand more:
http://www.khanacademy.org/math/algebra/linearequationsandinequalitie/graphing_solutions2/v/algebragraphinglines1
How are you supposed to find the fraction
aka the answer?! I need help.. any1?
aka the answer?! I need help.. any1?
IM THE SHORT GUY PEOPLE PLZ HELP!.... and thx and i wonder who else needs help ..
ya but your awsner doesn´t help :P
How do i turn x+2y=5 into slope intercept form? PLEASE HELP
TCulton,
The slope intercept form is y=mx+b
You have the form x+2y=5
In the slope intercept form, the y is all by itself on the left. So convert
x+2y=5 to a from that has y by itself on the left side of the equation
x+2y=5 First get rid of the x on the left by subtracting x from both sides.
xx+2y=5x so
2y=5x Now get rid of the 2 by dividing each side by 2
2y/2 = 5/2  x/2 so
y=5/2  x/2 Now change the order on the left
y=(1/2)x+5/2
So it is now in slope yintercept form. The slope is 1/2 and the yintercept is the point (0,5/2)
The slope intercept form is y=mx+b
You have the form x+2y=5
In the slope intercept form, the y is all by itself on the left. So convert
x+2y=5 to a from that has y by itself on the left side of the equation
x+2y=5 First get rid of the x on the left by subtracting x from both sides.
xx+2y=5x so
2y=5x Now get rid of the 2 by dividing each side by 2
2y/2 = 5/2  x/2 so
y=5/2  x/2 Now change the order on the left
y=(1/2)x+5/2
So it is now in slope yintercept form. The slope is 1/2 and the yintercept is the point (0,5/2)
Subtract x from both sides
Divide both sides by 2
Done
Divide both sides by 2
Done
At 7:20 why did he choose the second point, what if the first piont??
It doesn't matter which point you select. Sal probably picked the 2nd point because the zero usually makes some of the math easier to do. If you were to use the 1st point to find "b", you would get the same value for "b". Here it is:
5 = 5/6(3) + b
5 = 15/6 + b
5 = 5/2 + b
10/2  5/2 = b
5/2 = b
5 = 5/6(3) + b
5 = 15/6 + b
5 = 5/2 + b
10/2  5/2 = b
5/2 = b
how do i make 6x  3y = 9 in slope intercept form
Isolate y:
6x  3y = 9, add 3y to both sides
6x = 9 + 3y, then add 9 to both sides
6x + 9 = 3y, then divide everything by 3
2x + 3 = y
you can always enter this kind of stuff into wolfram alpha and they will solve it and graph it for you: http://www.wolframalpha.com/
6x  3y = 9, add 3y to both sides
6x = 9 + 3y, then add 9 to both sides
6x + 9 = 3y, then divide everything by 3
2x + 3 = y
you can always enter this kind of stuff into wolfram alpha and they will solve it and graph it for you: http://www.wolframalpha.com/
how do i put 6x  3y in slop intercept form
how do i solve 6x  3y = 9 for y
check my previous answer, but be aware that the value of y depends on x. Every solution is a point that is on the line 6x  3y = 9
You need to isolate y. Right now, y has a coefficient of (3) and 6x is added to it. First, move 6x to the other side of the equation by subtracting 6x from both sides (because we're doing the same thing to both sides we're not logically changing anything, we're only rearranging).
```
6x  3y  6x = 9  6x
```
Our equation simplifies to
```
3y = 9  6x
```
because
```
6x  6x = 0
```
and we can do addition/subtraction in any order we like so it doesn't matter that they're not right next to each other). We now need to remove the coefficient of 3. Note that 3*(1/3) = 1. So if we multiply both sides by 1/3, we'll be left with just y on the left hand side.
```
y = (9  6x) * (1/3)
```
I put parentheses around the subtraction of the RHS(right hand side) because we were multiplying the entire RHS by 1/3 so it needs to be distributed over addition/subtraction terms.
```y = (9 * (1/3)  6x * (1/3))```
becomes
```y = 3  2x```
and subtracting by a negative gives us addition so the final answer is
```y = 3 + 2x```
```
6x  3y  6x = 9  6x
```
Our equation simplifies to
```
3y = 9  6x
```
because
```
6x  6x = 0
```
and we can do addition/subtraction in any order we like so it doesn't matter that they're not right next to each other). We now need to remove the coefficient of 3. Note that 3*(1/3) = 1. So if we multiply both sides by 1/3, we'll be left with just y on the left hand side.
```
y = (9  6x) * (1/3)
```
I put parentheses around the subtraction of the RHS(right hand side) because we were multiplying the entire RHS by 1/3 so it needs to be distributed over addition/subtraction terms.
```y = (9 * (1/3)  6x * (1/3))```
becomes
```y = 3  2x```
and subtracting by a negative gives us addition so the final answer is
```y = 3 + 2x```
How would you put a problem like 2x  5y = 15 into Slope intercept form?
To be in slope intercept form, you need "y" to be by itself on one side.
So, in your example, move the 2x (subtract it from both sides:
5y = 2x + 15
Now, you need to make 5y into only y, so divide entire equation by 5:
y = 2x/5  3
So, in your example, move the 2x (subtract it from both sides:
5y = 2x + 15
Now, you need to make 5y into only y, so divide entire equation by 5:
y = 2x/5  3
i dont understand why negatives cancel out
A negative is the opposite of a number. The opposite of 2 is 2. The opposite of 2 is 2. Two negatives means 'the opposite of the neg. number', which is positive. Therefore 2=2. Three negatives (2) is the opposite of the opposite of the opposite of 2.
If you are not a boy, you are a girl. If you are not not a boy, you are a boy.
The thing that throws me off about slopeintercept form is that when I see the y variable, my brain wants to make that the yaxis. Does any one else find this confusing? Why is it like this? Is it because the coordinates are also written (x,y)?
y is a variable, and most often (at this level of math) it is the _dependent_ (sometimes also called the _output_) variable, and it _*is*_ associated with the y axis. Given a value of x (the _independent_ or _input_) variable, what pops out is y. The ordered pair of coordinates (x, y) can be thought of as (input, output) or (independent, dependent). So when you see (1, 5) what that means is there is a function, or expression, that when 1 is input, that is, when x=1, then what is output is 5, or y=5.
I assure you that you will grow accustomed to this representation over time. It is very common, and very useful.
I assure you that you will grow accustomed to this representation over time. It is very common, and very useful.
you know that why intercept is always (0,y) what ever the value of y is it will always intercept the yaxis. at (0,y).
the slope is the changey/changex which will give you how steep the line is. given the y intercept value in the equation, how y will move is dependent on how x moves. and you can derive that from the slope values.
Eg, y = 3/2x + 6
y inter is 6 (we know that because if you change the value of x to 0, 3/2 * 0 = 0, and 0+6 = 6)
so its (0,6) and slope values are 3/2 which means when x moves 2, y moves 3 (remember that y is dependent on x).
So x moved 2 (2+0 = 2) and y will move 9 (6+3 = 9). so the slope will be (0,6) (2,9) its going up by a ratio of 3:2.
the slope is the changey/changex which will give you how steep the line is. given the y intercept value in the equation, how y will move is dependent on how x moves. and you can derive that from the slope values.
Eg, y = 3/2x + 6
y inter is 6 (we know that because if you change the value of x to 0, 3/2 * 0 = 0, and 0+6 = 6)
so its (0,6) and slope values are 3/2 which means when x moves 2, y moves 3 (remember that y is dependent on x).
So x moved 2 (2+0 = 2) and y will move 9 (6+3 = 9). so the slope will be (0,6) (2,9) its going up by a ratio of 3:2.
Why does he subtract the two points when one is greater in x and y values than the other?
Humza,
For some reason, I can't access the video right now, but what Sal was probably trying to do was to calculate the slope of the line from two points that are on the line. In order to do that, the formula is m=(y2y1)/(x2x1). Basically that means that you call one of the points, "point 1" and the other "point 2". Then you subtract point 1's y value from point 2's, and divide that by point 2's x value minus point 1's x value.
For some reason, I can't access the video right now, but what Sal was probably trying to do was to calculate the slope of the line from two points that are on the line. In order to do that, the formula is m=(y2y1)/(x2x1). Basically that means that you call one of the points, "point 1" and the other "point 2". Then you subtract point 1's y value from point 2's, and divide that by point 2's x value minus point 1's x value.
The slope is the distance between y's over the distance between x's. You can start with either point first and you always subtract the two x values and the two y values starting with the same point for both x and y.
For instance, if you had two points (1,2) and (3,6) you could find the slope using either
(26)/(13) or (62)/(31)
They solve to:
(4)/(2) OR (4)/(2)
which both equal 2.
Does that help?
For instance, if you had two points (1,2) and (3,6) you could find the slope using either
(26)/(13) or (62)/(31)
They solve to:
(4)/(2) OR (4)/(2)
which both equal 2.
Does that help?
i cant figure out what (34,87) (51,25) can u plaese help me
I know that the slope is the change in y divide by the change in x. But what are you getting, if anything, if you divided the change in x by the change in y?
George,
If you take the *negative* of change in x over change in y, you would have the slope of a line perpendicular to the original line.
This video explains why: https://www.khanacademy.org/math/algebra/linearequationsandinequalitie/moreanalyticgeometry/v/perpendicularlines
If you take the *negative* of change in x over change in y, you would have the slope of a line perpendicular to the original line.
This video explains why: https://www.khanacademy.org/math/algebra/linearequationsandinequalitie/moreanalyticgeometry/v/perpendicularlines
I thought it was y2  y1 over x2  x1.
and it is, the short way to write it like you did is just to put the little triangle before the x and the y. tha just means change!
At 3:28 why does he put the first coordinate first? the equation is y>2 y>1
x>2 x>1
x>2 x>1
You can pick either one to be honest. you can start with the first coordinates or the second coordinates it doesn't matter the order.
Hi Sirena
From Mitchell
From Mitchell
he says change in y but the way i learned is rise over run so which one is it
Those mean the same thing. "Rise" corresponds to "change in y", and "run" corresponds to "change in x". See, if we denote "change in y" by ∆y and "change in x" by ∆x, "rise over run" refers to ∆y / ∆x, which is the slope.
how would you determinr the the rate of change is constant for tables given.
Spencer,
If the
(change in y) / (change in x) is always the same, then the rate of change is constant.
If you table has the following
(3,5)
(4,7)
(6,11)
(7,14)
You could calculate (change in y)/(change in x)
For the first two terms (75)/43) = 2/1 = 2
The second & third term (117)/(64) = 4/2 = 2
So far the change is constant
The third and fourth term (1411)/(76) = 3/1 = 3
This last point in our table shows we have a different ratio for
(change in y)/(change in x) so the table does not have a constant rate of change.
I hope that helps make it click for you.
If the
(change in y) / (change in x) is always the same, then the rate of change is constant.
If you table has the following
(3,5)
(4,7)
(6,11)
(7,14)
You could calculate (change in y)/(change in x)
For the first two terms (75)/43) = 2/1 = 2
The second & third term (117)/(64) = 4/2 = 2
So far the change is constant
The third and fourth term (1411)/(76) = 3/1 = 3
This last point in our table shows we have a different ratio for
(change in y)/(change in x) so the table does not have a constant rate of change.
I hope that helps make it click for you.
what does f(x)  f of x mean?
f=function. If you have an equation like y = 3x + 7 then you can also say that y is a function of x, or f(x). f(x) = 3x +7.
Sal, you kept saying that the points listed MUST satisfy the equation, what if the points do not satisfy the equation, I'm just wondering?
Points that dont' satisfy the equation cannot be on the line.
If a point does not satisfy the equation, the point cannot be on the line, therefore being unnecessary.
Sorry, could someone explain to me why you multiply x by the slope?
When x increases by 1, y increases by the slope m.
Consider the points (0,b) and (x,y).
Here the first coordinate increases by x and therefore the second coordinate should increase by m times x:
y = m*x + b
Consider the points (0,b) and (x,y).
Here the first coordinate increases by x and therefore the second coordinate should increase by m times x:
y = m*x + b
When finding the slope from the points could you just stack and subtract? Would that work everytime for points?
that was the way i was taught. it never gave me any trouble before. that math is much easier to do in your head anyway and maybe only need to write one or two numbers down till you get the answer.
what if all you have are two points? how would you find the yintercept? I still don't understand!!
Two points define a line, and if you define a line, then you define the yintercept. If you have two points, you can calculate the slope of a line between them. Then you can write y = mx + b, and you know x. Plug in the y and x of one point, and solve for b. That's the y interecept.
Please explain how to graph the equation and state the slope of the line if the slope existed Example x=8
x = 8 is a vertical line, because it says no matter what the y value is, x will always be 8.
The slope can be said to be "infinite", but since infinity is not a number, the slope in this case is "undefined", and that's what you should put down when taking a test.
The graph, once again, is just a vertical line extending in both directions forever, at x = 8.
Hope I helped!
The slope can be said to be "infinite", but since infinity is not a number, the slope in this case is "undefined", and that's what you should put down when taking a test.
The graph, once again, is just a vertical line extending in both directions forever, at x = 8.
Hope I helped!