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

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The discussion focuses on solving a quadratic system defined by the equations \( ax + by = 4 \), \( ax^2 + by^2 = 2 \), and \( ax^3 + by^3 = -1 \). Participants aim to find the expression \( (2x-1)(2y-1) \) based on the given conditions. The problem involves manipulating these equations to derive the values of \( x \) and \( y \) in terms of \( a \) and \( b \). The solution requires a systematic approach to isolate variables and apply algebraic techniques.

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

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

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

Neat
 

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