MHB Solving a System of Equations in N to Find a,b,c

[Albert1]

$a,b,c,\in N$
$( a −b)(b − c )(c + a ) = −90$
$( a − b)(b + c)(c − a ) = 42$
$( a + b)(b − c )(c − a ) = −60$
$ find :\,\, a,b,c$

 

[kaliprasad]

$a,b,c,\in N$
$( a −b)(b − c )(c + a ) = −90$
$( a − b)(b + c)(c − a ) = 42$
$( a + b)(b − c )(c − a ) = −60$
$ find :\,\, a,b,c$

a = 3, b= 1, c= 6

for solution
it is not elegent but effecitive

from the ( a − b)(b + c)(c − a ) = 42
as 7 devides RHS and not the other 2 equations we have
b+ c = 7 ( as b and c > 0)

so we get (a-b)(c-a) = 6

so combinations a-b = 6 , c- a = 1
a- b= 3 , c - a = 2
a - b = 2 c - a = 3
a- b = 1 , c- a = 6
and (-6, -1),(-3,-2), (-2,-3) and (-1,6) each can be tried with b+ c to give
a= 3, b = 1 and then they are seen to satisfy other 2 equations

 

[Opalg]

$a,b,c,\in N$
$( a −b)(b − c )(c + a ) = −90$
$( a − b)(b + c)(c − a ) = 42$
$( a + b)(b − c )(c − a ) = −60$
$ find :\,\, a,b,c$

My solution, like http://www.mathhelpboards.com/members/kaliprasad/'s, involved a bit of guesswork. Add the first and second equations to get $$-2(a-b)^2c = -48.$$ Add the second and third equations to get $$-2(c-a)^2b = -18.$$ Add the first and third equations to get $$-2(b-c)^2a = -150.$$ You can write those equations as $$(a-b)^2c = 24 = 2^2*6,$$ $$(c-a)^2b = 9 = 3^2*1,$$ $$(b-c)^2a = 75 = 5^2*3.$$ If you then make the assumption that the squared terms on the left side of those equations correspond to the squared terms on the right side, and the nonsquared terms likewise correspond, then you read off the solution $a=3,\,b=1,\,c=6,$ which turns out to be correct.

 

[kaliprasad]

My solution, like http://www.mathhelpboards.com/members/kaliprasad/'s, involved a bit of guesswork. Add the first and second equations to get $$-2(a-b)^2c = -48.$$ Add the second and third equations to get $$-2(c-a)^2b = -18.$$ Add the first and third equations to get $$-2(b-c)^2a = -150.$$ You can write those equations as $$(a-b)^2c = 24 = 2^2*6,$$ $$(c-a)^2b = 9 = 3^2*1,$$ $$(b-c)^2a = 75 = 5^2*3.$$ If you then make the assumption that the squared terms on the left side of those equations correspond to the squared terms on the right side, and the nonsquared terms likewise correspond, then you read off the solution $a=3,\,b=1,\,c=6,$ which turns out to be correct.

we can combine (from above)
(a−b)^2c=24=2^2∗6
and b+ c = 7 ( from by solution) and confirm c = 6
we could have (a-b) = 1 and c = 24 from the above and by trial and error remove it.
and combining the two we get c =6 ad confirm and then b= 1 and a = 3

combining the above 2 get less range to guess