MHB System of Equations: (2x-1)(2y-1)

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The discussion focuses on solving a system of equations involving variables a, b, x, and y, specifically ax + by = 4, ax² + by² = -3, and ax³ + by³ = -3. The values for a and b are derived as a = (4y + 3)/(x(y - x)) and b = (-4x - 3)/(y(y - x)). Substituting these into the third equation leads to the expression 4xy + 3x + 3y - 3 = 0. The conclusion drawn is that the best deduced form for (2x - 1)(2y - 1) is 4 - 5(x + y), which is seen as unsatisfactory. An alternative expression involving (2x + 3/2)(2y + 3/2) yields a more conclusive result of 21/4.
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If a\;,b\;,x\;,y satisfy \begin{Bmatrix} ax+by=4\\\\ <br /> ax^2+by^2=-3 \\\\ <br /> ax^3+by^3=-3 \\\\ <br /> \end{Bmatrix}. Then (2x-1)(2y-1)=
 
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jacks said:
If a\;,b\;,x\;,y satisfy \begin{Bmatrix} ax+by=4\\\\ <br /> ax^2+by^2=-3 \\\\ <br /> ax^3+by^3=-3 \\\\ <br /> \end{Bmatrix}. Then (2x-1)(2y-1)=
Solve the first two equations for $a$ and $b$. You should get $a = \dfrac{4y+3}{x(y-x)}$, $b = \dfrac{-4x-3}{y(y-x)}.$ Now substitute those values of $a$ and $b$ into the third equation. That should lead to $4xy+3x+3y-3=0.$

At that point I run into trouble. The best that you can deduce about $(2x-1)(2y-1)$ from that last equation is that $(2x-1)(2y-1) = 4-5(x+y)$, which does not seem like a satisfactory answer. If instead the question had asked for $\bigl(2x+\frac32\bigr)\bigl(2y+\frac32\bigr)$ then you could have re-written $4xy+3x+3y-3$ as $\bigl(2x+\frac32\bigr)\bigl(2y+\frac32\bigr) -\frac{21}4$ to give the answer $21/4$, which somehow seems to be more like the sort of conclusion that the question calls for.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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