The substitution method | Systems of equations | 8th grade | Khan Academy

The substitution method | Systems of equations | 8th grade | Khan Academy


Use substitution to
solve for x and y. And we have a system
of equations here. The first equation is
2y is equal to x plus 7. And the second equation here
is x is equal to y minus 4. So what we want to do,
when they say substitution, what we want to do
is substitute one of the variables
with an expression so that we have an equation
and only one variable. And then we can solve for it. Let me show you what
I’m talking about. So let me rewrite
this first equation. 2y is equal to x plus 7. And we have the second
equation over here, that x is equal to y minus 4. So if we’re looking
for an x and a y that satisfies both constraints,
well we could say, well look, at the x and y have to
satisfy both constraints, both of these constraints
have to be true. So x must be equal to y minus 4. So anywhere in this top
equation where we see an x, anywhere we see an
x, we say well look, that x by the second constraint
has to be equal to y minus 4. So everywhere we see an
x, we can substitute it with a y minus 4. So let’s do that. So if we substitute y minus
4 for x in this top equation, the top equation becomes 2y
is equal to instead of an x, the second constraint
tells us that x needs to be equal to y minus 4. So instead of an x,
we’ll write a y minus 4, and then we have a plus 7. All I did here is I
substituted y minus 4 for x. The second constraint tells
us that we need to do it. y minus 4 needs to be
equal to x or x needs to be equal to y minus 4. The value here is now
we have an equation, one equation with one variable. We can just solve for y. So we get 2y is equal to y, and
then we have minus 4 plus 7. So y plus 3. We can subtract y from both
sides of this equation. The left hand side,
2y minus y is just y. y is equal to– these
cancel out. y is equal to 3. And then we could go
back and substitute into either of these
equations to solve for x. This is easier right over
here, so let’s substitute right over here. x needs
to be equal to y minus 4. So we could say that x
is equal to 3 minus 4 which is equal to negative 1. So the solution to this system
is x is equal to negative 1 and y is equal to 3. And you can verify that it
works in this top equation right over here. 2 times 3 is 6 which is indeed
equal to negative 1 plus 7. Now I want to show you that
over here we substituted– we had an expression that,
or we had an equation, that explicitly solved for x. So we were able to
substitute the x’s. What I want to show
you is we could have done it the
other way around. We could have solved for y and
then substituted for the y’s. So let’s do that. And we could have substituted
from one constraint into the other
constraint or vice versa. Either way, we would have
gotten the same exact answer. So instead of saying x
is equal to y minus 4, in that second
equation, if we add 4 to both sides of this equation,
we get x plus 4 is equal to y. This and this is the
exact same constraint. I just added 4 to
both sides of this to get this
constraint over here. And now since we’ve solved
this equation explicitly for y, we can use the first
constraint, the first equation. And everywhere
where we see a y, we can substitute it with x plus 4. So it’s 2 times–
instead of 2 times y, we can write 2 times x plus 4. 2 times x plus 4 is
equal to x plus 7. We can distribute this 2. So we get 2x plus 8
is equal to x plus 7. We can subtract x from both
sides of this equation. And then we can subtract 8 from
both sides of this equation, subtract 8. The left hand side,
that cancels out. We’re just left with an x. On the right hand
side, that cancels out, and we are left
with a negative 1. And then we can
substitute back over here we have y is equal to x plus 4,
or so y is equal to negative 1 plus 4 which is equal to 3. So once again, we
got the same answer even though this time we
substituted for y instead of substituting for x. Hopefully you found
that interesting.

97 thoughts on “The substitution method | Systems of equations | 8th grade | Khan Academy

  • March 12, 2011 at 8:05 pm
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    what software was used for this presentation? I'm a college instructor, so i'm very interested ^_^

    Reply
  • March 14, 2011 at 6:19 am
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    @impdok
    MS Paint, really basic stuff. He also uses Camtasia Recorder ($200) + a Wacom Bamboo Tablet ($80) on a PC.

    Reply
  • March 28, 2011 at 6:08 pm
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    well…when you got 2y=3, didn't you hae to make it so theres only a variable…so divide 3 by 2?

    Reply
  • July 20, 2011 at 3:53 pm
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    WELL

    Reply
  • November 18, 2011 at 5:20 am
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    @MrDollerbill CONFUSING

    Reply
  • December 1, 2011 at 1:18 am
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    what if both parts of the problem equal y?

    Reply
  • April 19, 2012 at 5:09 am
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    wow u r a really good explainer. mhm that is osm sauce.

    Reply
  • April 29, 2012 at 4:45 pm
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    God …when my teacher explained it i could not understand 1 word, but you guys, i just wanna say thanks for this dudes. y'all ROCK!!!

    Reply
  • May 7, 2012 at 8:14 am
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    You are awesome…i finally understand it. Yay!

    Reply
  • September 18, 2012 at 1:17 pm
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    Thanks a bunch, this is great study for cramming 😉

    Reply
  • December 14, 2012 at 2:47 am
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    I can finally understand!!!!! thank you!!!!

    Reply
  • December 17, 2012 at 12:38 am
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    That's it? WOWWW

    Reply
  • December 18, 2012 at 6:41 pm
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    Say you have a problem like that, and it's a multiple choice question and it gives you answers to choose from. Would plugging each option in work too?

    Reply
  • January 19, 2013 at 12:00 pm
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    Name of the program pleasee !))

    Reply
  • January 28, 2013 at 12:42 am
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    thank you so much for this video!!!!! yay! now im not going to fail my test!!!!

    Reply
  • February 26, 2013 at 2:41 am
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    good tutorial

    Reply
  • April 18, 2013 at 3:55 am
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    I've been learning this for a month; this is the first time it's made sense in a reasonable way. thank you.

    Reply
  • April 22, 2013 at 9:14 am
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    thank you
    You made my hope that i can pass my math exam tomorrow

    Reply
  • May 2, 2013 at 8:53 pm
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    You teach better than my teacher

    Reply
  • May 27, 2013 at 1:42 am
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    Who else is here because they rage quit against their teachers

    Reply
  • May 29, 2013 at 1:47 am
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    I know we all are having problems with our math but when we think why are we learning this? We learn this to solve our life related problems. While, most of the people like us don't use this method b'coz we think it's hard.

    Reply
  • June 13, 2013 at 4:24 am
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    McGraw hall is starting to copy your idea their tutorials are not as good and schools are required to use them

    Reply
  • August 15, 2013 at 5:44 am
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    math is much more than "using the pythagoras theorem" or when i grow up when am i going to use this, its all about logic and critical thinking. which is why many people value math including universities and for people to say why am i learning math its not like im going to use it i want to be a writer – i mean yeah sure you're not going to use it firsthand but you develop higher thinking skills through the process of math

    Reply
  • January 2, 2014 at 10:19 pm
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    … Now i know my teacher i terrible. You make it look so easy.

    Reply
  • February 7, 2014 at 6:23 am
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    i didn't really understand it

    Reply
  • February 25, 2014 at 7:32 am
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    Really good stuff from what I heard in this video. But maybe take in mind the people who are colour blind and can't see some of this stuff properly…

    Reply
  • March 19, 2014 at 2:24 am
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    You teach soooo much better than my teacher!! My teacher just gives us worksheets for a week and expects us to understand and i learned more from a 4 minute video than in a week in her class

    Reply
  • April 23, 2014 at 1:05 pm
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    Thank you so much. Don't know what would have happen if you weren't on YouTube
    😀

    Reply
  • May 1, 2014 at 6:25 pm
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    THANK YOU SO MUCH

    Reply
  • December 10, 2014 at 1:07 am
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    but all my math H.W. problems don't have anything equal x or Y. It's just a number and it's really confusing. But thanks for making it easier to understand for the most part.

    Reply
  • December 12, 2014 at 1:06 am
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    i had harder versions of these 

    Reply
  • February 23, 2015 at 12:09 am
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    My teacher shows us how to do it ONE time then examples for a few days then new subject.

    Reply
  • April 2, 2015 at 1:40 am
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    This makes me hate my teacher even more. Thanks man! Great video!

    Reply
  • May 21, 2015 at 3:15 am
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    Thanks. My teacher sucks at explaining and a 4 minute free video helped me better than SCHOOL ITSELF.

    Reply
  • May 21, 2015 at 8:53 am
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    THANK YOU

    I was stuck trying to do my math homework, and this video beautifully explained everything so thaaaank you so much<3

    Reply
  • October 23, 2015 at 12:11 am
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    When are we going to use this in real  life????????????????????? like Fr

    Reply
  • November 28, 2015 at 4:41 am
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    Well as you can see the answer is 3 so 4 minus 7 is 3 you only have to do is to put only 3 no solution lol.

    Reply
  • January 6, 2016 at 1:07 am
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    this didn't help that much, you should add different types of problem to give the viewers a clue of how to solve them in my opinion

    Reply
  • January 20, 2016 at 3:25 am
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    I never listen to my teacher in class lmao, this why I'm looking up videos

    Reply
  • February 4, 2016 at 2:05 am
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    Yeah I'm a junior in high school here

    Reply
  • March 27, 2016 at 4:05 am
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    How you get 3

    Reply
  • March 28, 2016 at 6:19 pm
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    so all substitution have an answer

    Reply
  • May 26, 2016 at 8:46 am
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    im a sped help

    Reply
  • June 13, 2016 at 5:28 pm
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    When your teacher doesn't even teach you for your exam and tells you to go on videos instead

    Reply
  • June 13, 2016 at 5:31 pm
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    When your teacher doesn't even teach you for your exam and tells you to go on videos instead

    Reply
  • July 11, 2016 at 1:39 am
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    Nobody told me I could just -x when theres 2x I always tried to devide by 2 this makes my life 100% simpler

    Reply
  • August 4, 2016 at 1:35 am
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    what did they use to record video

    Reply
  • August 8, 2016 at 3:37 pm
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    My problems are much more complicated than this

    Reply
  • August 11, 2016 at 10:17 am
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    Thank you so much

    Reply
  • October 8, 2016 at 12:05 pm
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    Cool editting

    Reply
  • November 18, 2016 at 10:30 pm
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    easy money! I wish I can play game in class!

    Reply
  • December 12, 2016 at 5:37 am
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    I don't derstand where you got the 3 from

    Reply
  • December 30, 2016 at 12:54 am
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    if you get rid of the "y" at the end of the problem solving then you do NOT get y = 3; you get 2 = 3. That makes absolutely no sense. if you get rid of y, you have to bring down the 2.

    Reply
  • January 12, 2017 at 11:32 am
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    i understand this more than what my teacher discusses, thanks!

    Reply
  • January 17, 2017 at 11:19 pm
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    Please get a better microphone

    Reply
  • February 7, 2017 at 7:08 am
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    i watched this 1million times and i still don't get it
    ?…….

    Reply
  • February 22, 2017 at 11:15 pm
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    Now I know y math is so hard…..get it?

    Reply
  • June 15, 2017 at 2:51 pm
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    cheers helped me understand better

    Reply
  • September 20, 2017 at 3:47 pm
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    i honestly feel like an idiot for being the only one in class who doesn't get this

    Reply
  • October 19, 2017 at 10:20 pm
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    uh, im learning this in 10th grade. also, school didnt even teach this well

    Reply
  • November 1, 2017 at 2:14 am
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    Damn this was helpful

    Reply
  • November 8, 2017 at 3:08 am
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    I learned more in a 4 minute video than in a week of class. My math teacher is Khan.

    Reply
  • November 28, 2017 at 1:44 am
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    You need to do word problems this topic is still confuseing to me😬

    Reply
  • December 1, 2017 at 12:31 pm
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    I’m in 8th grade learning sophomore math😭 Thank you I finally understand it now

    Reply
  • December 2, 2017 at 6:46 pm
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    Despite watching this i still dont get it!

    Reply
  • December 8, 2017 at 4:01 pm
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    U are supposed to multiply seven times the x right not like this

    Reply
  • December 11, 2017 at 2:27 am
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    My accelerated math teacher just confused everyone in the class… I’ll have to show this to my friends so we’re not lost anymore 😁.

    Reply
  • December 11, 2017 at 4:10 am
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    I love you ! I was like how tf do I do this

    Reply
  • December 11, 2017 at 2:42 pm
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    succ

    Reply
  • January 7, 2018 at 7:55 pm
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    this is good for my final exam review

    Reply
  • January 17, 2018 at 1:21 am
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    this doesn’t make sense

    Reply
  • January 31, 2018 at 11:21 pm
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    where did the -1 come from

    Reply
  • February 3, 2018 at 11:57 pm
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    Saved me thank you

    Reply
  • February 7, 2018 at 2:16 am
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    What about a problem like
    Y=-2x+13
    X=9-y

    Reply
  • February 18, 2018 at 11:20 pm
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    THANK YOU SO MUCH 🙂

    Reply
  • February 26, 2018 at 3:45 am
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    6 years late but thanks

    Reply
  • March 9, 2018 at 3:40 am
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    I was so lost in my math class thanks I understand now

    Reply
  • April 4, 2018 at 10:23 am
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    Thanks so much teacher

    Reply
  • April 11, 2018 at 3:01 am
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    I thought this was hard in year 2… Now I'm thinking, where was khan academy back then???

    Reply
  • September 9, 2018 at 12:07 am
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    Im don’t understand this

    Reply
  • October 12, 2018 at 4:22 pm
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    Simple questions.. no one ever goes over the hard ones. Awesome

    Reply
  • October 24, 2018 at 3:02 am
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    Yeah um can you like do more problems then one please

    Reply
  • December 4, 2018 at 3:39 pm
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    My teacher gets mad at me for not understanding

    Reply
  • December 7, 2018 at 4:45 pm
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    THANK GOD FOR KHAN ACADEMY

    Reply
  • December 14, 2018 at 2:49 am
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    nani

    Reply
  • December 16, 2018 at 10:26 pm
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    Anyone know how he got -1 at2:29

    Reply
  • December 19, 2018 at 7:56 am
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    He lost me at y+3 someone tell me where it came from. He had me till he said that

    Reply
  • December 19, 2018 at 8:08 pm
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    In math class lol

    Reply
  • January 18, 2019 at 12:25 am
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    Anyone here Attention Deficit and can't pay attention, like, at all? :')

    Reply
  • February 12, 2019 at 8:54 pm
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    I knew that already just tell me how to do y=5x+6 y=6x+10 or any other duos like this one for subsitutuion

    Reply
  • February 21, 2019 at 3:01 am
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    Can someone help my question is -4x + 5y = 40
    X = 2y – 13

    Reply
  • March 6, 2019 at 9:12 pm
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    Don't UNDERSTAND!!!!!
    😣😣😣

    Reply
  • April 17, 2019 at 4:17 pm
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    Thank you khan academy.

    Reply
  • May 1, 2019 at 9:06 pm
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    Thank you, i had no idea what the teacher was saying at school but watching this video I understand now!

    Reply
  • January 14, 2020 at 4:07 am
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    What program are you using to write the equations on?

    Reply

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