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Systems of Equations

Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.

Factorization

         

If complex numbers \(x\) and \(y\) satisfy the simultaneous equations \[\begin{align} x^2+y^2 &= 4 \\ 2x^2-3y^2 &= -67, \end{align}\] what is the value of \((y+x)(y-x)\)?

Let \(a\) and \(b\) be the values of non-zero real numbers \(x\) and \(y,\) respectively, that satisfy \[3x^2+18y-12x=-9, x^2+9y-7x=-6.\] What is the value of \(a^2+b^2?\)

Let \(x=A\) and \(y=B\) be the solutions of the simultaneous equations \[\begin{align} \frac{3}{2x-1}+\frac{59}{y+1} &= 2 \\ \frac{3}{2x-1}-\frac{59}{y+1} &= 1. \end{align}\] What is the value of \(\frac{B}{A}\)?

If \(x=a \; (< 0)\) and \(y=b\) are solutions of the simultaneous equations \[2x^2-2y^2+3x-56y=44,\, x^2-y^2+x-29y=22, \] what is the value of \(b-a\)?

Let \(x=\alpha\), \(y=\beta\) and \(z=\gamma\) be the solutions of the simultaneous equations \[\begin{align} x+y-19xy &= 0 \\ y+z-12yz &= 0 \\ z+x-21zx &= 0, \end{align} \] where \( xyz \neq 0\). If the value of \(\alpha+\beta+\gamma\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is \(a+b\)?

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