Find all possible integer solutions that satisfy this system of equations:
a+b+c=1
a3+b3+c2=1
Let a+b+c=1, a^3+b^3+c^2=1. By inspection, b=-a is a solution.
a^3+b^3+(1-a-b)^2=1, or a^3 + a^2 + 2 ab - 2a + b^3 + b^2 - 2b = 0, so a^3 + a^2 + 2 ab + b^3 + b^2 = 2(a+b)
But : a^3+a^2+2ab+b^3+b^2 = (a + b) (a + a^2 + b - a b + b^2),
Thus (a+b) (a+a^2+b-ab+b^2)=2(a+b), (a+a^2+b-ab+b^2)=2.
Integer solutions with {a,b} unequal = {-3,-2}{-2,-3}{-2,0}{0,-2}{0,1}{1,0}.
Check for c with a+b+c=1: {-3,-2}{c=6}{-2,-3}{c=6}{-2,0}{c=3}{0,-2}{c=3}{0,1}{c=0}{1,0}{c=0}, all true.
So the solutions are {a,b,c} = {-3,-2,6}{-2,-3, 6}{-2,0,3}{0,-2,3}{0,1,0}{1,0,0}, {a=-b}
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Posted by broll
on 2023-10-01 08:40:10 |
Possible Solution
