# Equations of Lines and Graphing Hello again. So another 15 minutes on slopes
and equations of lines. We just got done talking about how slope is rise over run. That is
y sub 2 minus y sub 1 over x sub 2 minus x sub 1. We have some forms of equations of
lines. We have got Slope Intercept Form which is y equals mx plus b. We have Point Slope
Form which requires you to take a point and slope, the point is (X1,Y1) and the slope
is m. This is where we plug in our numbers when we find the equation of the line. We
also have Standard Form which is Ax plus By plus C equals 0. This is not very friendly
for graphing unless you are using the x and the y intercepts. We are going to pick back
up with some more examples on how to come up with some equations of lines given just
a minimum of information. The Point Slope form of a line tells us the absolute minimum
we need to write the equation of a line, a point x and y, and a slope. That is why it
asks for that information. In my last example we had… We were given two points, we found
the slope. We had to find our own slope and then we used the Point Slope form to find
the equation of the line. We are going to go right to that this time. Let’s say that
we have a slope of 2 and the point this time is not just going to be like (5,7). The point
is given to us as an x intercept. So my x intercept is equal to 4. So when I have a
point, I need an x and a y. But yet I said the x intercept is 4. Well, where is the x
and y? If you are crossing the x axis at 4… 1, 2, 3, 4… What is the coordinate of that
point? Well the coordinate of that point is 1, 2, 3, (4,0). So when I say the x intercept
if four, it only seems like you are given one number. I am actually giving you two.
You do have the x and the y coordinate. So below here we have a point again of (4,0).
That is going to be my x and my y. We are going to find the equation of the line. The
equation of the line. The Point Slope form says y-y1 so y minus the y value of the coordinate
is 0, equals m that is the slope I was given of 2, times x minus x sub one which is 4.
A lot of textbooks, and I won’t be happy if my kids give me an equation in this form,
because other than memorizing the Point Slope formula they have not done anything. So we
like to have these… I like to have these set equal to y. Because they are also very
easy to graph when y is by itself. So I am going to get rid of these parenthesis. We
are going to do that by taking the 2 and multiplying through. Subtracting zero does not do anything,
so I am not going to show that any more. y equals two times one is 2x, two times negative
four is -8. And BAM! I got my Slope Intercept form. y=mx+b So in case you are not very good
about graphing lines, let’s do that. This is my slope of 2 over 1. Now I am writing
this like a fraction because you need a rise value and a run value. Rise is either up or
it is down. It is up if it is positive, and mine is. And the bottom number I always read
graphs left to right. That is exactly a necessity, but I like to keep taking my up and down movement
as my positive movement or my negative movement. The bottom number always to the right. The
last number over here is my y intercept, my b. So b is equal to negative eight. When you
are graphing a line in slope intercept form, you do the y intercept first… I almost said
the wrong thing. You do the y intercept first of negative eight and then you do what the
slope says. Let’s see if we can graph it and get it to pass through this point like the
information said in the first place. It is in blue, so I will keep it in blue. The y
intercept is negative eight. Remember the y axis is the one that goes straight up and
down. So 1, 2, 3, 4, 5, 6, 7, 8. Now my slope is 2 over 1. So up 2 over 1, up 2 over 1,
up 2 over 1, and there is no reason for you to do this many repetitions. But look, I just
counted my way back to the point I was given in the first place. Pretty cool. So here is
my line. Y=2x-8. I have time for one more. Let’s take a look at… Lets do a weird one.
Lets say that my slope was undefined and the point that I am going through is… I don’t
know… (-2,4) Wow! I just told you that to find the equation of a line you needed a point
and a slope. I have a point, but my slope is undefined? How am I supposed to plug into
this equation… How am I supposed to plug into the Point Slope form if I don’t have
a slope? If you watched the last video, I kind of made some arm movements here trying
to show you some stuff. If you have a line going up to the right, that is a positive
slope. If it is falling down to the right, that is a negative slope. Horizontal lines,
that is a slope of zero. If you have a vertical line straight up and down, those lines have
an undefined slope. Unless you have that memorized there is nothing you can do with this question.
You are not going to be able to answer it. This means that you have a vertical line.
Vertical lines have the equation of x equals a number. In this case the x of the point
that the line is going through is negative two, so the equation is x=-2. That is it.
There are no x’s and y’s going on because the line is very simple. It is straight up.
The only thing changing is the y. I don’t have to know them. The only thing that I have
to know is that the x is staying a constant x value of negative two. If that is a special
case and it is pure memorization, what if I had a slope of zero? Well, if I have a slope
of zero and you don’t know the definition you can still work your way through the algebra
and get the right answer. But, if you know that… Ok, this is a slope of zero. Horizontal
lines have a slope of zero. Well, horizontal lines don’t have the equation x equals a number.
Horizontal lines have the equation y equals a number. So, y equals a number is the form
that we have for horizontal lines and what is the y value of this point that I am going
through. So take that number symbol out, y=4 and there is your answer. So there is two
examples. One in the old video and one in this one about finding equations of lines.
You need a minimum of a point and a slope. But there are these special cases. If you
don’t have these memorized, you are going to get those wrong. Let’s see here. I have
a few minutes left. Let’s just one more time take a look at graphing a line. We are going
to graph y equals 2x. Well slope intercept is y equals mx plus b. I have got the y by
itself so it is in slope intercept form. But it is… Oh. Well there is mx. There is nothing
over here. A lot of us will know what that means. But if you are just starting out in
Algebra you might not realize that there are telling you that the y value, the y intercept
excuse me, is zero. We don’t have to write plus zero because what is 5 plus 0? It is
still 5. It does not do anything. Well we have a slope of… You know what, let me make
this a negative. Negative two-thirds so yo can see that movement. So this line has a
slope which is negative 2/3. Now here is where I like to talk about your slope. It is negative.
I like to say that the top is rise up or a drop down. And this is going to go down because
the slope is negative. I like the bottom number is always to the right. Read those graphs
left to right. So we have a slope of negative two-thirds. We have a y intercept of b equals
zero. Again, one more time, when you are in slope intercept form you want to graph those
with the y intercept first and the slope second. So this first and this second. Here is your
line. Yes I do kind of write hard so I do break my chalk often. Y intercept is zero.
Done! Slope. Down two over three, down two over three. Or just to highlight it again,
down two over three to make my new point. There is my line. Let’s just say for a second
that this were y is greater than or equal to negative 2/3x. Now I have an inequality.
That means shading. Where to the y’s get bigger…greater than? This is y is greater than. Where to
y’s get bigger? When you go down? No, y’s get bigger as you go up. That is a y value
of 1, 2, 3, 4, so that inequality when you get to that in your Algebra, or Algebra 2
book, or Geometry will indicate that you shade up. Thank you very much. BAM!!! I am going
to BAM right back. Because I just realized that have a few more minutes to show you some
standard form? Well standard form is good if you are going to graph equations of lines
using x and y intercepts. Let’s talk about that. So here is an equation, 4x+3y+12=0.
I want a graph using x and y intercepts. If I don’t know what is happening, I am going
to erase this in a second, what is happening when you cross…when the line crosses the
x and y axis? If you are on the y axis, isn’t the x equal to zero. If I move from the origin,
this is up two. This point is (0,2). When you have a y intercept the x is zero. This
point over here that is on the x axis is (3,0). Three…Zero…so if I am on the x axis they
y value is zero. So y intercept, x equals zero. X intercept, y equals zero. I am going
to quicken this up just a little bit. I am going to bring the 12 over with subtraction.
4x+3y=-12. We are going to find the x intercept. You do that by letting the y equal zero. These
will fall away very quickly and you will have your answers. 4x plus 3 times 0 equals -12.
That is zero. 4x=-12. x when you divide both sides by four is negative three. Let’s find
one more point because you need two points to graph a line. My y intercept is when you
let x equal zero. That is going to be 4 times 0 plus 3y equals 12. I am almost out of space.
I am going to remind you that 4 times 0 is 0, and we are left with 3y=-12. When you divide
both sides by 3, -12 divided by 3…. I am undoing that multiplication… is going to
be negative four. So now I have two points. I have an x intercept and a y intercept. And
if I have two points I can graph a line. So can you:) We have an x y axis. We have an
x intercept of negative three. We have a y intercept of negative four. And through those
points we have a line. Now I am out of time. Thank you very much!!!

### 56 thoughts on “Equations of Lines and Graphing”

• August 22, 2012 at 12:03 am

How do you find the equation of a line given slope and a y-intercept? Also, thank you for making these videos, they're really refreshing my memory!

• August 22, 2012 at 12:05 am

How do you find the equation of a line given slope and a y-intercept? Also, thank you for making these videos, they're really refreshing my memory! Oh, never mind! I should have watched the second part before asking! Thanks again!

• August 22, 2012 at 1:41 am

HAHA:) No problem. Thank you very much for watching. Sorry for making you try a post a comment three times! I am a full time teacher and my students know about my YouTube channel too…of course. Because of that, and I don't want any potential problems at work, I approve any comments before they are posted.

• August 22, 2012 at 7:06 am

That's fine! I completely understand, and you're welcome!

• January 25, 2013 at 8:06 am

How do you do?

Can I graph a line -2X+1, by changing it into -2/1x+1?

• January 25, 2013 at 11:17 am

Yes…it is always a good idea to rewrite your slope as a fraction to you can see the rise/run vales when you graph:)

• April 23, 2013 at 2:55 am

i wish you were my math teacher, you made all the calculation so easy to comprehend and thank you for all of your videos. God Bless You and again thank you (:

• April 23, 2013 at 10:49 am

Now that you've subscribed I can be your teacher whenever you feel the need to watch and learn something new:) Thank you too for liking so many of my videos and I hope you will share my channel and your experience with your friends…my channel groWs through the support of my viewers:D

• April 25, 2013 at 1:34 am

You are an excellent tutor!! Instead of hiring a tutor for my *Intermediate Algebra" class, I now just go to your YouTube channel, and watch your videos. Since I have been watching your videos, my Intermediate Algebra grade has gone from a D- to a B+. Thank you so much for putting your algebra videos here on YouTube.

• April 30, 2013 at 9:42 am

WOW…thanks for sharing that awesome news with that glowing testimonial! And thank you to for supporting by subscribing:) I hope you will continue to share your experience and my channel info with others who might benefit as you did:D (I apologize for taking so long to reply…you got lost on the list temporarily:(

• July 28, 2013 at 6:28 pm

thank you very very much and sir you teach with concept wise and i am really thankful to
god that i have got you on youtube for teaching memaths with a very very great and brillant
teacher thank you once again sir

• July 30, 2013 at 2:25 pm

I'm just happy that my videos are helping you so much to keep you watching and learning! Thanks for always taking the time to thank me but you really don't have to…it's thanks enough that you have chosen my channel to learn from!

• August 13, 2013 at 10:15 am

fab!

• August 13, 2013 at 10:26 am

Look at you watching more videos already!…wooohooo!!!

• October 8, 2013 at 6:56 am

Thx a lot Sir, the concepts on lines and slope are crystal clear.

• October 8, 2013 at 10:42 am

You're welcome…thanks for watching!

• October 28, 2013 at 7:29 pm

I have a career where I have homework every night like my students do!

• October 28, 2013 at 7:30 pm

I don't have a lot of videos on vectors. You can find the ones I have by using the little magnifying glass under the banner on the homepage of my channel.

• December 9, 2013 at 3:13 am

Could you do a video where you have the slope but no b?
ex. y=-1/4x

• December 26, 2013 at 6:18 am

I'm in my final year of Mechanical Engineering and I've just managed to understand equations of line.. now everything i studied in the past makes sense.. Thanks Professor

• January 7, 2014 at 11:57 pm

I'm really sturggling in math right now, But I'm so glad that you're here to help. Thanks!

• February 24, 2014 at 3:08 pm

This guy performs truly an invaluable service, and for free!! thanks Prof Rob!

• March 11, 2014 at 10:37 pm

Thanku sir ! It really helped. 🙂
U certainly are doing a great job (y) 😀
BAM !!!!

• May 22, 2014 at 3:02 am

Very well explained!

• May 22, 2014 at 3:04 am

Thank you for sharing! Soon you will be collecting the fruit of your a m a z i n g work!!

• June 11, 2014 at 12:30 am

Very good at stating the little things.  This video has highlighted one chapter of a book I am studying.  I had some terrible little questions that have been answered here.  I am really glad I found this video.

• June 25, 2014 at 1:04 am

Thank you for all this, now I understand graphics and they don't seem confusing. You're cool, man.

• July 16, 2014 at 11:38 pm

beautiful

• September 6, 2014 at 10:31 pm

Great Work !! You are a lifesaver!!

• September 9, 2014 at 2:41 am

Awesome video, great for refreshing for Calc.

• September 19, 2014 at 4:27 pm

I can't thank you enough for making these videos.  I am over 60 years old and very grateful for all the extra help you are providing for my PreCalculus studies.  It's like having a tutor at home every day.  Another wonderful lesson.  Thank again!

• October 23, 2014 at 5:24 am

You sir are amazing. I was having such a hard time trying to understanding point slopes and different formulas. Not only did you explain it really well, you also gave out very useful examples with thoughtful explanations step by step. And because of that you just got yourself a sub. Keep up your outstanding work and I like how you have enthusiastic attitude towards your audience while you teach compare to other teachers who bores everyone. Again thank you so much for your great lesson you put up there 🙂

• November 4, 2014 at 7:29 am

You are a top dogg man!! So sarcastic and funny!!

• January 21, 2015 at 4:13 pm

Just to let you know, I received my AA degree in December 2014, graduation ceremony will be this coming May. Thanks again for all your help.

• January 29, 2015 at 12:30 am

I love it!

• February 26, 2015 at 2:02 pm

Hats off to you sir always optimistic and trying to give the Best. My university is going to start in august and thanks to your videos I'm gonna rock (BAM!!) calculus. You are the best .

• March 15, 2015 at 10:19 am

i'm not sure if this has to do with anything, but what if the equation is " y = x-3 " i have  this maths test tmrw, and when i tried to do this in my book, the answer said something else…

• March 24, 2015 at 3:47 pm

THANK YOU AGAIN 🙂 🙂

• April 12, 2015 at 1:06 pm

Thanks for these great videos sir.

• April 23, 2015 at 8:33 am

I'm not sure this video is for who but I'm in my last year of high school and I'm revising everything with the help of your video. It really helps me. So, thank you. Your videos are great !!!

• April 27, 2015 at 12:40 pm

This video and the video after it in the geometry playlist are in the wrong order. In the next video you link to this video as being part 2.

• June 27, 2015 at 8:37 pm

YOU ARE AWESOME ROB! THANK YOU!

• July 3, 2015 at 5:06 am

You really do make the best straight lines without a ruler! I'm going into pre cal and i just wanted to know what i'm going into. After just finishing trig the videos of this play list so far seem like a review of algebra 1 with the slope stuff and i kno i did good with that so that won't be hard to remember. Is trig going to come back into pre cal? will we be doing a lot of those trig proofs again? (which i know were your absolute favorite of all math! lol) thanks again i just wanna make sure i'm prepared!

• August 21, 2015 at 4:49 pm

How can you draw a line that straight?! 4:56

• August 24, 2015 at 12:24 am

I love that you are using a real chalkboard! Rockin it old school! Thanks!

• December 10, 2015 at 3:13 am

Isn't standard form Ax+Bx=C instead of Ax+Bx=0 because the last one is in general form?

• February 15, 2016 at 4:58 pm

• February 15, 2016 at 5:06 pm

i have been having trouble with this but you got me to understand this thank you

• May 1, 2016 at 12:23 am

Thank you so much for making these videos! Yours are one of the only videos that explain the math in a way that makes sense to me! Thank you!!

• September 5, 2016 at 8:19 pm

You should talk to Hank Green about making some Crash Course math series.

• September 26, 2016 at 2:21 pm

Do you have somekinda stencil or other gadget to make that coordinate grid on your chalkboard? I first thought it was permanent until you erased it.

• April 19, 2017 at 5:50 pm

Bam Bam Bam! Right into my head. Thanks😃😃

• September 28, 2017 at 8:08 am

wow That WAS bam! ☆☆☆Bam × 4…and wait…there's more.
love it👨🙆💕🏩🍁🌱🔥

• October 1, 2017 at 3:57 am

You are amazing! Those who have given you a thumbs down are probably those who are here trying to sell tutoring services. hahaha.

• October 15, 2017 at 7:20 am
• 