Math Olympiad Question | Learn How to Solve the System of Equations Fast | Math Olympiad Preparation

Learn how to solve the Olympiad Question quickly with these tips and tricks for X, Y, and Z variables. The system of equation involves X, Y, and Z variables.

Today I will teach you tips and tricks to solve the given olympiad math question in a simple and easy way. Learn how to prepare for Math Olympiad fast!

Need help with solving this Math Olympiad Question? You're in the right place!

I have over 20 years of experience teaching Mathematics at American schools, colleges, and universities. Learn more about me at

Math Olympiad Question | Learn How to Solve the System of Equations Fast | Math Olympiad Preparation

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[haofengxd]

Hello sir, long time no see. I am so busy these days, so i don't have time to watch your videos. But so good to see you again! Nice problem and could you make more videos about Geometry? Thanks so much! I wish nothing, but the best for you!

[Copernicusfreud]

I solved for y with the first two equations. y = x/(5x-1) and y = z/(6z-1), If y = y, then x/(5x-1) = z/(6z-1). Solving for x with that equation gives x = z/(1 - z). The third equation in the problem is z + x = 7xz, and solving for x in that equation gives x = -z/(1 - 7z). If x = x, then z/(1 - z) = -z/(1 - 7z), which eventually produces 8z = 2 and z = 2/8 or z = 1/4. Plugging z into other equations gives x = 1/3 and y = 1/2.

[anatoliy]

Thank you so much, Professor! Along with your lessons, I seem to be reliving my childhood. And your Math Olympiads Preparation are well reminiscent of my preparations for the School Chemistry Olympiads. Then I twice went to the republican level and both times won prizes. God bless you!

[pedroloures]

I think you missed a solution, (x, y, z)=(0, 0, 0). When you divide by XY you're assuming that they're not zero! But thanks for the problem!

[johnwindisch]

The magic of substitution! Still learning when to apply it. Solved it a harder way.
Thanks

[shadmanhasan]

Thanks for the challenge Premath. Always lookin forward to a nice way of thinking 👍

[HappyFamilyOnline]

Very nice explanation👍
Thank you so much for your hard work 😊

[xyz]

your first step was assuming none of them is 0, but x, y, z all equal to 0 is a valid answer. If one of them is 0, from the equations, it will make all of the them 0, so we can take the alternative that none of them is 0, and follow your steps to get the 2nd set of answers.

[math]

you handled this Olympiad question very well with your tips and tricks, excellent presentation bro

[devondevon]

1/2, 1/3, 1/4 Answers
a different approach
x+y=5xy equation 1 hence
hence y= 5xy-x
y= x(5y-1)
y/x = 5y-1 equation 4

y + z =6yz
y =6yz-z
y = z (6y-1)
y/z =6y-1 equation 8

from equation 4:
-1 = y/x-5y AND
from equation 8:
-1 = y/z-6y
since both equal -1 then
y/x-5y= y/z-6y
-5y+6y =y/z-y/x
y = y/z-y/x
y = y (1/z - 1/x)
1 = 1/z - 1/x
xz = x-z (multiply both sides by xz)
But it is given that x+ z = 7xz (from the third equation in the video). so add this to xz=x-z

x+z=7xz
x-z = xz

2x = 8xz
1= 4z hence
z =1/4 one answer
since y + z = 6yz then
1/4 + z = 6y(1/4)
y + 1/4 =3/2y
1/4 = 1/2y
1/2 =y multiply both sides by 2 another answer

Since x+ y = 5xy given then
x+ 1/2 = 5x(1/2)
x +0.5 = 2.5x
0.5 = 1.5x
0.5/1.5 =x
1/3 = x
Answer x =1/3 y =1/2 and z =1/4

[rachidbenmeziane]

nice and simple, good job, really I like it

[philipkudrna]

Before watching: x=1/3, y=1/2 and z=1/4. (Or alternatively, all are zero.) I did a traditional approach, expressing y in terms of x from the first equation and inserting it into the second and so on. Probably not the most efficient way, but in worked. Now I watch!
Ok, my terms were a bit more cumbersome, but I think my approach was even faster or at least not slower! 😃

[alster]

After seeing the

2(a+b+c)= 18 technique, I paused the video and solved it on my own


a+b+c=9

Same drill. After getting the final answer, I fast forwarded to the end to double check.

[kailashanand]

goodevening sir, there is one more solution you missed because of the initial division, (x, y, z) = (0, 0, 0) also works!

[Skank_and_Gutterboy]

Nice problem. It's not too often that I do the problem, then watch the video to find out that I did it just like you and got the right answer. One thing is for sure: since I started working the problems on this channel, I have become a lot more comfortable with using substitution as a viable method and I can see that it's actually a viable method, not just some hack or trick that is better to avoid. Through my years of school, unfortunately substitution was never taught very well and the instructors tended to treat it as a hack or trick, too.

[mathsdone]

Beautiful video with nice explanation. Thanks sir 👍👍

[tetsujin]

I did it slightly different.
x-7xz = -z -> x(1-7z)=-z -> x = -z/(1-7z)
y-6yz = -z -> y(1-6z)=-z -> y = -z/(1-6z)
Substitute in the values of x and y in x+y=5xy and solve for z, then work out x and y

[soniamariadasilveira]

Obrigada, querido professor, o senhor nos faz parecer tudo muito fácil com suas explicações.

[arthurschwieger]

You make this look so easy! I love your videos. :-D

[BrunswickTchatchou]

Good morning and thanks for all you do🙏🙏🙏🙏