MHB System of Equations: (2x-1)(2y-1)

AI Thread Summary
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.
juantheron
Messages
243
Reaction score
1
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)=
 
Mathematics news on Phys.org
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.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
3
Views
2K
Replies
16
Views
4K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
5
Views
2K
Back
Top