Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: √ ∞ ≠ ≤ ≥ ≈ ⇒ ∈ Δ θ ∴ ∑ ∫ π -

Login

Username

Password

Not registered yet?

#1 2006-04-07 00:28:15

krassi_holmz
Real Member

Offline

a b + c d = 6 && a c - 3 b d = 5

Find all integer solutions of the system:
a b + c d = 6 && a c - 3b d = 5.


IPBLE:  Increasing Performance By Lowering Expectations.

#2 2006-04-07 11:16:56

George,Y
Super Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

(a+d)(b+c) = 6+5+4bd


X'(y-Xβ)=0

#3 2006-04-07 12:02:30

mikau
Super Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

thats the solution?


A logarithm is just a misspelled algorithm.

#4 2006-04-07 18:18:09

krassi_holmz
Real Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

:0
No. You haveto find all fours (a0,b0,c0,d0) elem Z:
a0 b0 + c0 d0 = 6 && a0 c0 - 3 b0 d0 = 5


IPBLE:  Increasing Performance By Lowering Expectations.

#5 2006-04-08 01:13:47

George,Y
Super Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

(a+d)(b+c) = 6+5+4bd
contains all the information of a b + c d = 6 and a c - 3b d = 5. and it might be a path. i'm new in this field, and what i could do is shifting, to see if it has been made simplier....dunno

Last edited by George,Y (2006-04-08 01:15:54)


X'(y-Xβ)=0

#6 2006-04-08 03:33:42

krassi_holmz
Real Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

You are not so right:
{a b + c d = 6 && a c - 3b d = 5} => {(a+d)(b+c) = 11+4bd}, but:
{a b + c d = 6 && a c - 3b d = 5} !<=> {(a+d)(b+c) = 11+4bd}, so not all solutions of your equation will be solutions to the system.


IPBLE:  Increasing Performance By Lowering Expectations.

#7 2006-04-08 18:07:00

George,Y
Super Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

yes, you'r right. sufficient but not necessary. at least one of the original equation have to be contained.
from  a c - 3b d = 5, ac-5 contain 3 , ac  elem {8,11,14...}


X'(y-Xβ)=0

#8 2006-04-09 20:10:24

krassi_holmz
Real Member

Offline

Re: a b + c d = 6 && a c - 3 b d = 5

a c elem {...8,11,14,...}:: a IS integer


IPBLE:  Increasing Performance By Lowering Expectations.

Board footer

Powered by FluxBB