Multiple examples of constructing linear equations in slope-intercept form
Multiple examples of constructing linear equations in slope-intercept form
- In this video I'm going to do a bunch of examples of finding
- the equations of lines in slope-intercept 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 y-intercept.
- 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 y-intercept 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 y-intercept 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
- 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 y-intercept
- But we could figure out both of them from these
- 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 color-coded-- 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 color-coded 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 y-intercept.
- 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 y-intercept?
- Well here we can just look at the graph.
- It looks like it intersects y-axis 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 y-intercept.
- 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 y-coordinate.
- 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 y-values-- 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
- color-coding 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 y-intercept.
- 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 left-hand 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?
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=> y - y1 = mx - mx1
=> y = mx - mx1 + y1
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/point-slope-to-slope-intercept-form.php
y-y1=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=-1/2x-1....that is the slope intercept form
y-4/6/8/10/12 you have to find a pattern (in this case y is twice the amount of x) so y=2x
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
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.
slope = (5-0)/(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."_
Comment if you get it or if you meant something else
Why isn't it just 6/-3 ? I never understood this and it always stuffs me up.
Your mistake is that (6-0) 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 (6-0)/2-5)
(6-0) is +6 , not -6
(2-5) is -3
So the fraction is 6/-3
which reduces to -2/1 = -2
I hope that helps make 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 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
And the graph would be a vertical line running through the 3 on the x axis.
Hope that helps :-)
x-intercept 3, y-intercept 2/3?
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!
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.
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!
Hope that helps!
m is the slope and b is the y-intercept.
Can you plze explain how to write an equation with this example?
Also, the f at around 10:32 is a symbol for function.
"The line contains points (2,6) and (5,0)"
In this problem, Sal subtracts this way, "6-0/2-5"
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 "0-5/3 - -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,
Sal's Equation: "0-5/ 3- -3"
Hope that helps!
So, you have to get that equation into standard form and then see what's in front of the x.
-4x - 3y = 16
+4x........+4x......(the 4x's cancel out on the left side)....(the dots are for spacing)
(divide both sides by -3)
Notice '-4/3' is in front of the x, so it would be 'm' in y=mx+b, or the slope.
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/linear-equations-and-inequalitie/more-analytic-geometry/v/perpendicular-lines
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!
(change in y) / (change in x) is always the same, then the rate of change is constant.
If you table has the following
You could calculate (change in y)/(change in x)
For the first two terms (7-5)/4-3) = 2/1 = 2
The second & third term (11-7)/(6-4) = 4/2 = 2
So far the change is constant
The third and fourth term (14-11)/(7-6) = 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.
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=(y2-y1)/(x2-x1). 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 instance, if you had two points (1,2) and (3,6) you could find the slope using either
(2-6)/(1-3) or (6-2)/(3-1)
They solve to:
(-4)/(-2) OR (4)/(2)
which both equal 2.
Does that help?
-5x+6 has a slop of -5 (meaning that the line goes up 5 units in the y-direction while going left 1 unit in the x-direction) and the first point you would graph, starts at (0,6) which is 6 on the y-axis.
-(5x+6) however means the whole expression is being multiplied by -1. Which then means you would get -5x -6. This would mean you would have the same slope as the previous (up 5 units in the y-direction and 1 unit to the left in the x-direction) but it also means the first point to graph would be -6 instead of +6.
This would result in two lines which have the exact same slope (and therefore parallel) but are 12 units apart in the y-direction (meaning one starts out at +6 on the y-axis and the other starts out at -6 on the y-axis.
Hope this helps!
Because the y-intercept is 6, the line passes through the point (0,6).
To get a second point that the line passes through, start with any point on the line (the y-intercept will do) and add 1 to the x-value and add the slope to the y-value. So the second point will be (0+1, 6+(-5)) which is (1,1).
So we mark the points (0,6) and (1,1) on the graph and draw a straight line through them, and that's our graph of y = -5x + 6.
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!
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/
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.
And why is it always y=x+(number) or y=x-(number). Why not just have say, 3 coordinates? So say x,y,s. (s for slope).
And why does y=y/x+(number). I get it in terms of algebra but why do we solve it the way we do?
Line contain points (2,6) and (5,0)
but the formula is
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.
2y=5-x 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
So it is now in slope y-intercept form. The slope is -1/2 and the y-intercept is the point (0,5/2)
if it is right then please prove it to me.....thanx!
6x - 3y - 6x = -9 - 6x
Our equation simplifies to
-3y = -9 - 6x
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))```
```y = 3 - -2x```
and subtracting by a negative gives us addition so the final answer is
```y = 3 + 2x```
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:
aka the answer?! I need help.. any1?
I don't understand this at all please help
"The slop is undefined (some would say infinite), and there is no y intercept. So you can't write it in slope-intercept form".
The line is an up and down line, which never intersects with the Y axis instead it runs parallel at the point x= 3. So every point will have 3 as it's x and an infinate amount of answers for the y in it's coordinates. The x never changes. Therefor the equation for that line is simply "x=3" and there is no way to put it in slope intercept form, as there is no slope to the line it is just strait up and down.
Now if you had a line that was y=3 than, slope intercept for that line is y=3, as the slope of that line is 0. I know it's funny, but the best way to think about it is if you were to set a ball on the line, the line x=3 is unable to provide any guide for the ball, and it just falls. If you set the ball on the y=3 line it just sits there and does not move (supposing you are able to set it perfectly still).....least that's how I remembered....
Question- Line that goes through the point (1,5) and the slope is undefined
You first have to find the slope of the line which is change of y over change of x.
That would be 4-7/2--2 or -3/4.
After you find the slope, you can start forming the equation
You can plug the values of either of your known points (that is (2,4) and (-2/7)) and solve for b.
After a water main break, a large building's basement was flooded to the ceiling. The local fire department sends over 2 pump trucks. Truck No. 175 can pump 25 cubic feet per minute and truck No. 236 can pump 32 cubic feet of water per minute. The buidlings basement is a large rectangular prism measuring 150 ft long, 120 ft wide, 10 feet deep. If both pump trucks are used for different amounts of time to pump out the basement, define variables for the time that each truck pumps, and then write an equation the represents this situation. Help?
That just gets you the slope though, from there you need to use the slope in an equation that represents the line. The slope intercept form being used in this video is one of the most useful. In it you will place the slope you found in place of m in: y = mx + b. The b represents the y-intercept, or the value of y when x = 0.
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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
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This is great, I finally understand quadratic functions!
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