Solving Systems of Equations... Elimination Method (NancyPi)

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Published 2018-05-16

MIT grad shows how to use the elimination method to solve a system of linear equations (aka. simultaneous equations). To skip ahead: 1) For a BASIC example where terms cancel right away when you add the equations, skip to 0:25. 2) For an example in which you have to MULTIPLY ONE EQUATION by a number before adding the equations, skip to time 6:26. 3) For an example in which you have to MULTIPLY BOTH EQUATIONS by numbers before adding, skip to 12:12. P.S.) For HOW TO SUBTRACT equations instead of adding them, if you'd rather do that, skip to 18:00. Nancy formerly of MathBFF explains the steps.

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For how to solve a system of linear equations by the SUBSTITUTION METHOD, jump to: • Solving Systems of Equations... Subst...

What does it mean to solve a system of equations using the elimination method? It means to solve for the x and y values that make both equations true, using the elimination method. Elimination is just one way of solving (substitution is another way) and is sometimes called linear combination. Here's how:

1) BASIC ELIMINATION: First, make sure the equations are ordered in the same way, with their like terms lined up vertically. Then, try adding the equations (vertically) to make a new equation. If a term in the first equation has the same number coefficient but opposite sign (like 5y and -5y), then adding the equations will mean that those terms cancel. Since those terms disappear (in the example, 5y and -5y), there is just the variable x left in the equation so you can solve for x. Once you solve for x, you can take that number and plug it in for x in either of the original equations in the system, to solve for the other variable, y. The (x,y) pair is your solution.

2) MULTIPLY ONE EQUATION BEFORE ADDING: Sometimes terms don't immediately cancel when you add the two equations. What do you do then? Try multiplying one of the equations by a number (both the left and right sides) to make either the x terms or y terms have similar coefficients (but opposite sign), like positive 4x and negative 4x. Then add the two equations (the new version) and follow the same steps as in the first example.

3) MULTIPLY BOTH EQUATIONS BEFORE ADDING: If it does not help to just multiply one equation by a number, you may need to multiply both equations, separately, by numbers. If you check to see whether you can just multiply one equation by a number and add opposite terms/eliminate a variable, and it is not possible, then aim for an LCM, or least common multiple, by multiplying each equation by a different number, and then adding the equations.

P.S.) YES, YOU CAN SUBTRACT INSTEAD (if you want): If you have a system that has the exact same x terms or exact same y terms in both equations, if you prefer, you can just subtract the equations. Alternatively, if you still wanted to add the equations, you could just multiply one of the equations by -1 first and then add the equations and solve normally.

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 have 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: nancypi.com/

All Comments (21)

@maryannmonteverde5160 4 years ago

I have been a math teacher for 25 years. I am now using your videos to help my students adjust to distance learning. You do a great job, Nancy. Your voice is very calm and you demeanor is reassuring. Thank you!
@franchescanetto2311 4 years ago

I want her to be my teacher at my school. I never understand what my teachers are saying. But the way she explains it, it makes it so easy to learn.
@selena396 5 years ago

I’ve been trying to learn this for a week and somehow magically learn it in 5 minutes
@giovannicloete4588 5 years ago

wowwwwww learnt more in 20 mins with you than my last two years of school!!!!!!!!THANK YOUUUUUU
@aaronharris8093 2 years ago

I'm a freshman in college studying electrical engineering, and this video is a life saver! Thanks a million!
@chrismatthews1722 4 years ago

Absolutely excellent instruction: clear, concise, comprehensive, simple. Thank you SO much for what you are doing to encourage and give hope to so many of our children. It has been 30 years for me and even I picked this up like it was nothing!
@glabuser2176 4 years ago

Hottest math teacher ever. Not even gonna cap, you helped me complete my homework assignment. I appreciate it nice job 👍
@jrbleau 4 years ago

I'm a mathematician, I know this stuff like the back of my hand, yet still I watch it! Kudos Nancy!
@ginnystack368 5 years ago

This video makes it all make sense. You didn't just answer the question you explained all the possibilities. Thank you Thank you Thank you. You are a wonderful teacher.
@stephenugochukwu9472 4 years ago

thanks a lot Nancy for making out time to educate people like me, who really need help in maths.
@wyattlambert2947 4 years ago

Just me, or is her voice soothing?
@musicis4life917 4 years ago

wow so my during this whole quarantine my teacher posted this with a video that didn't help and most things algebra i'll understand...... if it's explained to me right but she went step-by-step and not only did she do more then one problem she didn't do the easiest one..... what I'm trying to say is she just saved me from ripping out my hair. THANKS SO MUCH, YOUR AMAZING!!!
@jannebukomah4816 4 years ago

Hey ! I just wanted to tell you Thanks for all your help, Your videos always help me so much easier than teachers would've been able to do for me in just a period of 20 minutes, I appreciate it !
@gregedwards9901 6 years ago

You are a great teacher.
@brandongonzalez6277 6 years ago

I cannot believe it's finally happened, and these videos feel just the same as the classic Nancy mthbff. So great to have you back, and I'm glad things have become better for you on the other side to have enabled your return. Wishing you the best - gratefully, a mentee.
@CaptainAMAZINGGG 4 years ago

I literally have forgotten basically ALL of math that I learned in high school. And tbh before boys happened, I did well in math, and LOVED algebra, I've just remembered. It's not really in my mind anymore, but I think this video helped me be RE-in love with it. :') Imma relearn the things now because of it. I actual miss knowing algebra.. What a life plot twist wtf hahaha. <3 <3 <3 Thank you so much :3 <3
@theinternetplumber1365 6 years ago

We're being netflixed and we love it. It's like you just dropped a season on us today. Thank you!
@kathrynhooper2351 5 years ago

This was so helpful. I had no idea how to do this as all the other videos were unclear and messy. Not only this they didn't use equations that needed either 1 or both sides to be multiplied. Thanks this was the last part I needed for my math exam tomorrow (technically today as I have been up all night into the morning studying... oops).
@Emerardo 4 years ago

Thank you! I've been struggling with this for a while, but you put it very simply and got straight to the point. Most helpful 20 minutes of my life lol
@ibrahimmosavi2168 2 years ago

I really enjoy her lectures. She makes complex concepts simple to understand.