# Solving Systems of Equations… Substitution Method (NancyPi)

308,806
Published on May 16, 2018 by

MIT grad shows how to use the substitution method to solve a system of linear equations (aka. simultaneous equations). To skip ahead: 1) For a BASIC SUBSTITUTION example, skip to time 0:19. 2) For an example where it's TRICKIER to solve and also involves FRACTIONS, skip to 08:17. and P.S.) For WEIRDER-LOOKING DIFFERENT FORMS of systems, for which you can still use substitution, skip to 16:46. Nancy formerly of MathBFF explains the steps.

For how to solve a system of linear equations with the ELIMINATION METHOD, jump to: https://youtu.be/XOJgzW4P7T8

What does it mean to solve a system of equations using the SUBSTITUTION METHOD? It means to solve for the x and y values that make both equations true, using the substitution method. Substitution is just one way of solving (elimination is another way). Here are the THREE STEPS you can always follow:

STEP 1) First, solve for x or y in either of the two equations in the system (get x or y alone on one side). It doesn't matter which equation you use first, and it doesn't matter whether you get x alone first or y alone first. You can pick whatever looks easiest to solve for in the system.

STEP 2) Then, take the expression you found for y (or x) and plug it into the other equation you haven't used yet. For instance, if you got y alone in the first step, take that expression for y and plug it into, or substitute it into, the variable y in the other original equation in the system. Then, simplify the expression by distributing to open up the parentheses (using the distributive property) and combining like terms. Solve for the unknown variable in this equation.

STEP 3) Finally, once you solve and get an actual number for a variable, you can take that number and plug it back into either original equation in the system and then simplify and solve for the other variable. This (x,y) pair is your solution.

IMPORTANT: If you get something like 0 = 0 or 8 = 8, or something like 1 = 2 or 0 = 21 when you are following these steps and trying to solve, then you've probably found a special case. If you get something TRUE like 0 = 0 (a number equals the same number), then there are an infinite number of solutions to the system, and you can just write "infinitely-many solutions". If instead you get something FALSE like 0 = 21, then there is no solution (inconsistent system), and you can write "no solution".

For more of my math videos, check out: http://nancypi.com

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• Nice way to explain
Thanks

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x times y ?

in the 3rd one last example……

• You make the equations easy to solve. Well spoken, clearly defined problems and solutions. My ERAU instructor linked us to this video, very pleased. Thank you!

• Which number did she not write in the video?

• On the second problem I paused it to try to practice elimination (reviewing lol). I multiplied the bottom equation by 2, cancelling out the x and y giving me 0=0 (instead of -8=-8.. i think if you simplified that it’s still 0=0, right?) Still infinitely many solutions, but is that still a correct way? Man I’ve been overthinking a lot if so lol thank you again!

• I did without simplifying by 2 and found 8 = 8. I do not know what this means. Nor did I continue the resolution.

• Jugginator 12, yes, the two equations are the same line.

• Ronald Jensen Thank you!

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Video, no liar

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I spent soo much time trying to learn this from my textbook but you just made it so easy. You deserve more subs!

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2/3x + y = 14/3
4/5x – 3/4y = -10
Thank you

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• 9:41 wait wouldnt you add the 2y to the 4 insted of tagging it at the end? can someone explain lol i have a quiz tommorow

• i guess i’m a bit late :), but 2y and 4 and are not like terms. this means they can’t be added together. this is because (y) could be anything and adding 2y and 4 could be 504 or it could be -7. this is why you have to put it in its simplest form; 2y+4. i hope this helped and you did well on your quiz

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• I remember linear algebra. That was some really bizarre but cool mathematics.
Have you ever set up a system of equations and then converted the matrix into echelon form obtaining the answers on the right side of the equals sign? I got fast at that once, but I think I’m a little out of practice. Once, I also wrote a very cool computer programming routine that worked like a charm in most situations. Imagine a program where all you have to do is put in the known variables, and it automatically solves for the unknowns. This is a science all of its own!

• Ψ΄Ψ±Ψ­ Ψ±ΩΨ­Ψ© Ψ΄ΩΨ±Ψ§Ψ§Ψ§Ψ§be love Nancy omyma form lraqπππ

• Nancy really needs to name the “tops” and the “bottoms” for what they actually are as numerator and denominator. We should not give in to the dumbing down of students. They need to learn the terms.

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4x+7y=-3

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I didn’t really understand what my teacher was saying in class and this video helped me understand.

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• can i ask something ; i know im dumb at math but.
im on the middle of my math solving and then
* -37x = 0 *
i had almost got the x , but then i dont know if it will be like this
* -37x Γ·37 = 0Γ·37 *
i know im dumb π

• I was going to recommend this video to my class, because your explanations are so organized and clear… But then you got to the comment about the system with infinitely many solutions, where you say that -8=-8 regardless of whatever values x and y take, so any pair (x,y) is a solution. π Not every pair would solve the system. Only those that are on the line.

I am looking for a video where students can learn that “infinitely many solutions” is not the final answer. Instead, they need to present the solutions as a dependent pair. In your example, the answer to the question of “solving the system” would be that there are infinitely many solutions, and they are of the form (4/3+2/3y,y), or (x,3/2x-4)

I would love it if you switched this video for one like that, because I have a feeling you would do a great job explaining it.

• Thanks for the video, it really helped me so much. Keep up the excellent work girl πͺ

• At the beginning of the the video what if the 2 was a negative? Would you add or subtract?

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• Nancy, you made a mistake solving the second equation. X is 2Y, and Y is 1. So there is no infinite solutions, there’s only ONE.

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• I love you continue teaching us how to solve the problem of equation you teach us with much love

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π

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• The elimination better would’ve been better/faster

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• Finnaly I simple way to do I found videos that legit took 20 minutes to answer 1 problem

• no wait but,i used -6x + 4y=-8 then i got y=2
but then u used “3x-2y=4 and then urs was infinitely many solutions” how come?

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• Go on sis, solve those equations.

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• Mam i really liked it I’m from india

• Hi Nancy! I found that this works a little easier and w/o dealing with fractions. multiply the bottom equation by 2 and u will see it works out very nicely. I always take the easiest term that will cancel out the top/bottom term to eliminate x/y. In this case u got the best of both. π

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• Finally, you are an angel sent from above by the math gods. Thank you very much..

• I m the student here… I never saw any teacher being nervous in front of listeners …. Good teaching ma’am

• For the 6x + 4y one, I got x=0 and y=-2. It works too! I’m very confused.

• In the second equation, she got x in terms of y. Then, when you take that expression (from the second equation) and plug it in for x (in the first equation), you wind up getting a true statement (anything=itself), after working out all of the arithmetic. When that occurs, there are infinitely many solutions to that equation.

• When i did part 2 by my self following the steps i was like what heck am i doing something wrong after trying it a few times i was like wow i am really messing up something so simple then i gave up and watched the rest of the video and a turns out its a special case that say/ means they can be infinitely many solutions or no solutions i honestly just learn that this could happen so glad i watch this video

• You saved my life!!!ππ

• thank you so much, this video helped me a lot but what is there were eqn. that stared with 1/2(m-3n)=1
3(m+4n)=-12

i have no idea what i should be doing after i get y=mx+b

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• who else is confused

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• hey, thank you so much. The video was great, i could solve the equations but cannot solve the questions based on word problems. The equation that i create is absolutely absurd and never correct. Could you please make more videos on how to solve word problems on simple equations, ages, reverse number and stuff .

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• X=2 and y=7 correct answer

• Hi
The video is cool, but please if u can solve ( 2x + 5y = 33 & ( x + 3y = 19 ) using substitution method that will be great. Thanks

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• x=4 ; y=5

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Math:iNfInItLeY mAnY sOlUtIoNs

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• How you write from your left hand

• How long and tedious this is is really confusing me. I have 2x+7y=222 and 2x+136=6y and I canβt find anything videos that are set up to help me with an problem like that. My jackass of a math teacher gave us homework the night before a final. Heβs a piece of shit.

• Ur an angel

• Thanks

• Life saver

• For some impressive reason, women most often seem to teach so much better than men. So many male teachers offer the easy examples. You instead chose to show the more challenging ones with fractions as well, offering a way to see the fractions’ solution when it is not obvious (as in whole number / one). You offer students a way to solve these with assurances that they too can get math done. Had I had your teaching when I was young, I might’ve even fancied myself as a modern-day Euclid. Well done, thank you.

• Very nice videos

• All of ur video is very helping me in my math. Last month, I got 86% for my mark. Thanks

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• After watching this video 10 times now I got it

Maybe…

• Mode of instruction made it much easier to understand. Thank you!

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• shes not writing backwards and shes not left handed, they flipped the video

• great explanation and a beautiful accent.

• I saw this video as was scrolling through the internet for systems of equations, why because I was failing my 7 grade algebra class I saw the video a good 3 times and I turned into a calculator. thank you so much. (T_*) thank you

• any videos on gaussian elimination

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• Confused

• Thank god, finally a pretty math teacher and the name Nancy pi perfect. God bless you Nancy.

• Her voice us calm

• Question: is it normal for a variable to be equal to a decimal? e.g. x=1.9 ?

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• Everyone here left handed (including me) hit the like button
π

• But 2x minus 2 is zero not -2

• writing backwards is haaaaaaard

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• lmao itβs flipped. sheβs not writing backwards

• with all regards,
i did the last one and have a little confusion on the signs Q) Y=-X+1
Y (place) Y=X-7
-X+1= X-7 (1 go’s to right hand side 7+1=8)
-X=X-8
-2X=-8 -8/2
(simplifying) -8/2=4 now here i am little stuck with signs its negative 8 over positive 2 now will i be getting -4 in results or +4
X=4 or X=-4
(Y=X-7 in place) (Y=X-7 in place)
Y= 4-7 = -3 OR Y=-4-7 =-11
PLEASE ANYONE WHO knows HOw To DO THIs help me

• Omg thank you girlπ­ I needed this!! Thank you a million times youβre a legend!!

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• my momsphone YEah Iβm uploading tomorrow π because Iβve been busy with college π

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• Where did you get that -4 ??

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Thanks a lot πππππ

• who else is watching this *full volume mid test?*

• Angela Mosby How?

• Wtf??π

• Γ-4y=3
2Γ+5y=3

• The Best teacher ever thank you Mam for this video

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• Reading backwards, writing backwards and writing lefty. Wow. Couldn’t do that in a million years.

• Thanks Nancy! you rock!

• Where were you when I was taking HS Algebra? You have a pleasant demeanor and your students should be grateful.

• you seem like ur being held at gunpoint, are u ok?

• Thank you for all the help

• Elimination would not have worked either for the last case because of signs of positive and negative not allowing for clean multiplication

• Damn, that was hard to just watch but I think I got it. Thanks so much for making these vids you might of just help me pass my finals will keep you updated if I pass ^ ^

• Did you pasd

• wait in 3:30 i don’t know why you subtracted the minus two x from the eleven, i thought that you could only take it away from the x
any help will be much appreciated
XD

• its pretty simple
the first eqn is 2x + y = 11 and you want to isolate for y
so you’d subtract 2x from both sides so it would be,
2x – 2x + y = 11 – 2x
y = 11 – 2x (2x-2x = 0 so it would 0 + y which is simply y)

• @Sanchiv Suresh glad i’m not the only one doing this π

• woah! this what i was looking for thanx this helped me a lot

• Thats so easy π

• 2Γ+y=you’re an angel on earth

• You do you write backwards

• TMR is my final and this is the only thing I struggle on.. so Iβm so thankful I found this video you teach better than my teacher<3

• I have a test in a hour…. I AM CRAMMING INFO IN NY BRAIN BECUZ I NEED DIS π

• Ur so pretty i cant focus in next viedio wear maskππππ

• What are you gonna do about the voice?

• Ur so good at teaching nancy i love math and also uπ

• Thank you so much Nancy. It’s so great to see a woman making such an impact in the math world. Beautifully chosen examples by the way. My class loved this video.

• Proof that the genetic lottery does exist! Congrats on your winnings Nancy! Incredibly helpful videos and I appreciate your time efforts.

• y’all telling me she was writing backwords?? NOW that is true commitment! ANd thx your helped me study for my finals thx thx thx.

• Thank you for the help, I am a new subscriber from now on!

• 2020 person watching a 2018 vid :]

• can you make English vids

• Smajh nhi aaya pr sun kr acha lga….

• Thanks so much for making me understand
I love itππ

• Thank you.

• How study without writing on your notebook

Me: watching on youtube on how tf to solve

• But with question two they are just the same function. You can see this if you multiply all terms by -2 in the 2nd function. That’s why there are infinite intersections and therefore infinite solutions.

• Is she writing in reverse direction?

• Best teacher on Youtube, I mean it.

• Great video

• Spoiled

• Aftet watching this video in class

I still failed.

• Thank you fancyPi

• I like you and your video baby

• Thank you Nancy! I know that this might be over 2 years too late but your videos really helped me out because later in the school year I will be taking my EOC!

• Can you like please be my math teacher mines a bitch π

• her students are blessed

• 14:34 the answers i got were
X=1.3
Y=-0.05
And they are correct because when you substitute them you get the correct answer! π€

• Im Filipino who hard to understand math and i think this video can helf me so much to understand ..

• U seem nervous idk why

• Thank you so much; you are a lifesaver!

• This is the only one that helped me

• hi there! do you also add graphing…

• thank you I am in eighth grade and I am using this!

• Really loved this it was so useful Thank you

• Great work!

• 10:53 Why canβt we simplify that???

• because it gets simplified later when you times it by 4. If you did it then you would get decimals which would make it way more confusing.

• You are so hot

• People will come by looking at you
Agree πππ

• can every youtuber/math teacher put their last name in the channel name, bc i need that for referencing and i cant find any other source ππππ

• U are sexy for that 1 like
We’re Indians we can understand any topic much faster than any country in this world
Sorry but it’s the reality
If u want to realize u can test our strength

• The COrona virus is upon us and I’m watching this video, it was really helpful! thanks!

• Are you in class 10th tell meπ

• thanks … keep it up …nancyπ

• I love the way you explain math problems. don’t change π

• i like the video but i also like that its a beautiful lady teaching me all my other maths teachers are males

• Honesty Nancy, if you have a school on your own. i bet you your school will be the top grade-A school ever, because one can really understand your teaching well.

• Cool maths thanx

• JosΓ© Miguel 1de abril 2020

• Can someone help me substitute this
6x 9+6y= -6
5x + y = -13

• tysm its hard to learn in online school and this is the only video that truly helped π

• thanks so much

• Thanks a lot!!

• nice

• Thank you so much for taking your time to help others this video helped me out so much. Now I feel I can be successful in college algebra at jersey college.

• Thank u for cleaning my doubt

• Write better backwards than I do forwards

• Thank you

• You’re too much!

• Thanks!!

• MIT Rocks!!!

• its confirmed ill never be good at math this is really hard

• this video has no sound for me

• you’re even better than Sal Khan

• If only i had a teacher like you i won’t be coming 4th…..i would be 1st in my class

• anyone know what -6=x/6 is??

• MALE ZABOUR

• Jesus is Lord! He will bring you peace! Repent! He is coming soon!

• That u

• This is brilliant I like this

• Thanks di

• Math: exists
My brain: *explosion*

• If we want to use the 2nd equation first we have to use same method

• Go to this channel

• Wow, an instructor that tackles the tough cases (which are always on the homework, but never in the examples). You instruction is a breath of fresh air!

• that long really! i can make it much more shorter and simple

• I already understand it the only reason why I watch the videos is because I have a crush on her

• yo whats ya number tho

• im not even paying attention im just amazed on how she can write pretty backwards and also trying to figure out what shes writing on maybe on glass? thin air? idk

• editing….

• I assume it a piece of glass and a mirror.

• So cute π

• U are so pretty and thanks

• It is really really helpful ππ

• Best

• Thank you! I have a test on friday and this helped. My teacher doesn’t really know how to use the online learning software.

• thanks nancy

• DAMN is she was my teacher I would definitely focus in this class

• If you want to understand the concept please visit the channel

• Thank you so much this really helped me!

• So my school isn’t doing the online classes, so I’ve been on Wattpad and Webtoon for the past 4 and a half months killing my brain cells (not that I’m complaining). Basically, I have to take it upon myself if I’m gonna learn, and it’s like your re-wiring my brain.
Thanks!

• Fun fact: your searching for this the day before the test

• Nahhhhh, I have my final in an hour. Yeah, I’m dead π

• @Qasim gl on ur final. Idk if ur seeing this rn cuz it says 1 hour ago but you know

• @xxxDea_ Thanks a lot, and yeah, it’s done and I think I probs passed, yayyy π

• i love the way you teach nancypi

• this video is too nice

• Thank alot

• Show partial fraction decomposition please!

• Thank you so much but i am still kinda confused..

• Helo nancy mam i am from india and i like your video very much….

• Recognize that its not easy job to write backward.u did it nice π

• Thanks, I finally learnt that π

• They just flip the video.

• Who else is watching this lockdown

• In first question at the final step why dont you take 15 to the other side and divide it with 30 it looks more easier and on the left hand side there will be only x so the result is same

• i love how she explains it really softly , it just feels soothing to hear and understand β€οΈ

• Nice explanation #NANCY

• im focusing on the teacherπ€£

• Really your explanation is simple and worth learning.

• Thx but I am speechless when I learned algebra and is too hard for me

• I am a 7th grade