MHB Solve Quadratic System for $(2x-1)(2y-1)$

[anemone]

For all real $a,\,b,\,x,\,y$ such that

$ax+by=4,\\ax^2+by^2=2,\\ax^3+by^3=-1.$

Find $(2x-1)(2y-1)$.

 

[Albert1]

For all real $a,\,b,\,x,\,y$ such that

$ax+by=4---(1),\\ax^2+by^2=2---(2),\\ax^3+by^3=-1---(3).$

Find $(2x-1)(2y-1)$.

Spoiler

$(2)\times x :ax^3+bxy^2=2x---(4)$
$(2)\times y :ax^2y+by^3=2y---(5)$
from $(1)(3)$ we get $(4)+(5)\rightarrow -1+4xy=2x+2y\rightarrow 4xy-2x-2y+1=1+1=2=(2x-1)(2y-1)$

 

[anemone]

Spoiler

$(2)\times x :ax^3+bxy^2=2x---(4)$
$(2)\times y :ax^2y+by^3=2y---(5)$
from $(1)(3)$ we get $(4)+(5)\rightarrow -1+4xy=2x+2y\rightarrow 4xy-2x-2y+1=1+1=2=(2x-1)(2y-1)$

Well done Albert, and thanks for participating!

 

[kaliprasad]

Spoiler

$(2)\times x :ax^3+bxy^2=2x---(4)$
$(2)\times y :ax^2y+by^3=2y---(5)$
from $(1)(3)$ we get $(4)+(5)\rightarrow -1+4xy=2x+2y\rightarrow 4xy-2x-2y+1=1+1=2=(2x-1)(2y-1)$

Spoiler

Neat