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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.

System of Equations - 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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