Hardest maths questions: simultaneous diophantine equations

This question from the 2007 Australian Mathematics Competition (senior division) involves 3 simultaneous equations with positive integer solutions. We have to find the smallest value that x can take. 0% of students got this one right! I'll be making heavy use of the elimination method in this video, although you could use substitution instead.

[rajeshmittal]

Yeah i got this one question also correct
Minimum values
X =83
Y =7
Z = 13
U = 17
Nice question
Many students will loose by seeing 4variables and 3 equations

Moral: Don t loose your hope
Never loose your confidence
Believe in yourself that you are the genius and to proove this do everyday a challenging problem

Luv you all
Luv you maam for teaching us

[AnirbanUpadhyaya]

I tackled this question by using the value of x from the first equation to eliminate it from the other 2 equations. This gave me 2 equations with 3 variables: y, z and u. I eliminated z to get the equation 7u=17y, which gives u/17=y/7, which I took =k for the general solution. Putting y=7k and u=17k in the previous equations gives me x=83k and z=13k.
I checked the original 3 equations to confirm this solution works.
And so, putting in k=1 gives us the minimum positive integer solution.

[ghislainmaury]

Good job,

But after having found that 13x = 83z, I wrote x = 83n (=> z = 13n) to check the value of u and z depending of n:
(after a first error in my solving with y not of the same sign of the others, I prefer to prove the value/sign of each variable depending of x)

with (4) => 4*83n - 14y -18*13n = 0 => y = 7n
with (1) => 83n + 7n - 3*13n -3u = 0 => u = 17n

Then: x = 83n, y = 7n, z=13n and u=17n
All numbers have the same sign
the minimum positive value of x = 83 (n=1)
then y=7, z=13 and u=17

(You said it with the good value, but it's still better to write it in an exam/competition, to prevent any error)

[shashankpandey]

Your problems are awesome and the solutions seems to be tough but are easy when you disclose the solutions

[anhstanislawski]

very interesting problem hope you post more videos
<3<3<3

[shashankpandey]

Hey pls help me get rid of this question - " if a+b+c= 63, what's the maximum possible value of ab+bc+ca? I did it almost half but couldn't manage to get the right one.

[Gamma_Digamma]

We could potentially row reduce couldn’t we ?

[blaze]

You’re so beautiful, both on the outside and inside! Keep up the great work 😍❤️.

[shreya]

Can you plz suggest some site where I can get past year papers of the Australian mathematics competition...I can't find them anywhere.

[shankarrahi]

mere pass pi ka current value hai mere pass
pi ka man 22/7, 3.14, 223/71, 355/113, root 10, bhi nahi hota hai
current value mere pass hai

[nellvincervantes]

How old are you miss? I like women older than me especially with brains like you. Dont know why I am attracted to them.