All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
3 unknowns (Posted on 2023-09-18) Difficulty: 3 of 5
The set of integers {x,y,z} complies with the following set of equations:

x+y+xy=17
x+z+xz=35
y+z+yz=71

Evaluate {x,y,z}.

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 3 of 4 |
Simon's Favorite Factoring Trick
(x+1)(y+1) = 18
(x+1)(z+1) = 36
(y+1)(y+1) = 72

let a=x+1, b=y+1, c=z+1

1) ab = 18 = 3*3*2
2) ac = 36 = 3*3*2*2
3) bc = 72 = 3*3*2*2*2
Since multiplying any two of these yields a positive number, 
a,b,c must either all be positive or all negative
From equations 1 and 2: c = 2b and 
          from 2 and 3: b = 2a; so c = 4a
a*2a = 2a^2 = 18, a = ±3
b = 2a = ±6
c = 4a = ±12
(a,b,c) = {(3,6,12) or (-3,-6,-12)}
Subtract one from each:
So (x,y,z) = {(2,5,11), (-4,-7,-13)}

{edited to correct an arithmetic error}

Edited on September 19, 2023, 10:00 am
  Posted by Larry on 2023-09-18 14:17:13

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (17)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information