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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 - Elimination

If $$x, y$$ are distinct values satisfying $$x^2 = 21x + 12 y$$ and $$y^2 = 21 y + 12 x$$, what is the value of $$x^2 + y^2$$?

What positive integer value of $$M$$ satisfies the following system of equations:

$\begin{cases} \sqrt{M} + \sqrt{N} = 25 \\ M - N = 175 ? \\ \end{cases}$

Let $$a$$ and $$b$$ be the values of real numbers $$x$$ and $$y$$, respectively, that satisfy $x^2-xy+y^2=3, x^2+y^2=5.$ What is the value of $$\lvert{a+b}\rvert?$$

Suppose $$x$$ and $$y$$ satisfy both of the nonlinear equations $\begin{cases} x^3 + y^3 & = 62 \\ x^2 - xy + y^2 & = 31. \end{cases}$ Find $$xy$$.

If $$x=a \; (> 0)$$ and $$y=b$$ are solutions of the simultaneous equations $9x^2-10xy+y^2=0,\, 5x^2-y^2=64,$ what is the value of $$a+b$$?

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