Linear equations and functions

# Slope-intercept form

## More examples of constructing linear equations in slope-intercept form

Linear Equations in Slope Intercept Form
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## More examples of constructing linear equations in slope-intercept form

Discussion and questions for this video
how do you change from point-slope form to slope-intercept?
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

1 Comment
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.
1 Comment
at 3:27 shoudnt it be y2-y1 over x2-x1 instead of y1-y2 over x1-x2
Believe it or not, they are the same thing. Try it.
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.
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!
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.
1 Comment
to find the slope sal subtracts (at 3:00) 6-0/2-5. check. next problem i paused, tried to figure it out myself and subtracted (at 5:54) 5-0/3-(-3) to find the slope. sal however reversed the order and did 0-5/-3-3. 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 = (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."_
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 y-intercept can be any number, it need not be a whole number (usually is not). You just plot 2.5 as best you can, half-way between 2 and 3.
This isn't exactly related to the topic, but why does 6-0/2-5 become -6/-3 ?
Why isn't it just 6/-3 ? I never understood this and it always stuffs me up.
Tomm0,
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.
1 Comment
How Would You Work This Type Of Problem When It's Set Up In A Table???
do you mean if its like
x-2/3/4/5/6
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
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.
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 y2-y1/x2-x1. So you would have 0-6/5-2. 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 6-0/2-5 or the way 0-6/5-2. You probably thought that 0-6 was just 6, but 0-6 is -6. Then you get 5-2 which is 3, so -6/3. If you thought 0-6 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
write th equation of the line in slope- intercept form if the slope is 1/5 and the y-intercept 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
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 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 y-intercept of the equation if it was not written in standard form ? 2 Votes 1 Comment 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. What is the point of graphs in life? What are they used for? Graphs are just a way to represent data in a way that people can easily understand. For instance, if the average person heard that "sales have risen an average of approximately 2.7% each year for the past 7 years, while costs have risen at a rate of approximately 3.2% per year", that might just sound like boring, senseless numbers. But with a graph, it becomes easier for people to understand what the data really means. For instance, if you looked at how high sales are on a graph, and how high costs are on a graph, it becomes more clear whether your business is succeeding or not. If the line for sales is high while the line for costs is low, you're doing well. Otherwise, you're in trouble. It's that simple. So, graphs are just a way to show the information in a way that's simple, clear, and easy to understand. And there are other ways to do this as well. Pie charts, tables, scatter plots, histograms, box-and-whisker plots, and so forth are all just methods of showing data without making it sound so complicated. And of course, there are mathematical advantages to graphs as well. Often times, the simplest way to solve a tough math problem on paper is to just graph it and look at the graph. You'll find this coming in handy more and more as you go into higher levels of math. Anyways, I hope this helps clear things up a bit. 2 Votes 1 Comment AT 2:23 I don't get how he got -x Try watching the video again. How do I write the slope-intercept 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 :-) 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 how do you find a y intercept in a graph You find the y intercept of a graph by first inputing everything that you know into the equation: y = mx + b 'm' would be the slope, and 'b' would be the y intercept. You find the slope by finding two plotted coordinate pairs and putting them into the equation y2 - y1 --------- x2 - x1 You then use one of the coordinate pairs and input the new numbers into the equation. Afterwards you then simplify until you find b. Ex. (3,2) and (2,1) y = mx + b y2 - y1 1 - 2 -1 --------- = ------- = ----- = 1 x2 - x1 2 - 3 -1 2 = -3 (same thing as 1x) + b 2 = 3 + b -3 -3 ----------- -1 = b y = x - 1 If you input one of the two coordinate pairs, the equation will work. 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. Is it m=y2-y1 over x2-x1 or m=y1-y2 over x1-x2? 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 y1-y2=-(y2-y1), annd the same for the x's, so multiply one way by -1/-1, and you'll get the other. 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! 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 How do you solve a problem when it asks you, " Write the equation that describes each line in slope-intercept 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 X2-X1. 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: 4-2 over 3-1 which is 2 over 2 which can be simplified to 1. The slope of the equation is 1. Hope this helped! Isn't -2-1=3 ? Sal said that it equals to 2?! The answer is not 2. Probably just a simple mistake 2 Votes 1 Comment why are putting 6-0 6-0 just goes with the concept so he put that there, Originaly it would be just 6, unless it was: 0 - 6 i hope this helped. :) what is the *slope-intercept formula* and x and y are points on the graph. i have this question: "write an equation for a linear function whose graph has the given characteristics... Passes through (3,0) parallel to the graph of g(x)= -2/3-4" ... NOT looking for the answer, but the correct video to learn how to work this and similar problems I couldn't find one specific to parallel lines, but here is one with perpendicular lines. The process is the same, except parallel lines have slopes that are the same. So, 1) Find the slope for g(x). It will also be the slope of the equation you are trying to create. 2) Use the slope and the point (3,0) to create the equation. Here's the link for the video: https://www.khanacademy.org/math/geometry/analytic-geometry-topic/parallel-and-perpendicular/v/find-the-equation-of-a-line 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)/(3--3) = 5/6 (0-5)/(-3-3) = -5/-6 = 5/6 Both give the same answer. 3 Votes 1 Comment At 6:11, how come its 0-5 instead of 5-0? Is that crucial to get the correct answer or does it not matter, because you are still finding the change? (y2 - y1)/(x2-x1) 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. 3 Votes 1 Comment 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?
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.
Will you please show how to find the slope-intercept form for the line satisfying the following:
x-intercept 3, y-intercept 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!
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.
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.
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 slope-intercept form, which is *y = xm + b*. "b" is the y-intercept 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/linear-equations-and-inequalitie/equation-of-a-line/v/graphing-a-line-in-slope-intercept-form.

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".
1 Comment
How do you write an equation in slope intercept form?
A linear equation in slope-intercept form is y = mx + b. Where "m" is your slope and "b" is your y-intercept.
How to graph y=x
The y-intercept 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=y1-y2/x1-x2

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.
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, "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,

"5-0/ -3-3"

Sal's Equation: "0-5/ 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!
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.
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.
1 Comment
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.
How do you find the y-intercept when you find the slope of a line in a linear equation?
When you get the slope plug it in to the following equation:
y=mx+b
Where y is the y point, and m is the x point. x is the slope you found earlier.
For example, say I have the point (4,5) on a line which has a slope of 3.
I would do: 5 = 4 (3) +b
-Distribute on the right side: 4 (3) = 12.
5 = 12 + b
Subtract 12 from the right side to get b = -7.
From this information you can write down the equation as y = 3x -7 OR y = 3x + -7 (does not matter which you do though).
How are you supposed to find the fraction
aka the answer?! I need help.. any1?
IM THE SHORT GUY PEOPLE PLZ HELP!.... and thx and i wonder who else needs 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.
x-x+2y=5-x so
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
y=(-1/2)x+5/2
So it is now in slope y-intercept form. The slope is -1/2 and the y-intercept is the point (0,5/2)
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
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/
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
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
1 Comment
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.
The thing that throws me off about slope-intercept form is that when I see the y variable, my brain wants to make that the y-axis. 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.
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=(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.
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.

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
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.
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.
1 Comment
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 (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.