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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 - Substitution

If $$x$$ and $$y$$ are non-zero real numbers satisfying the system of equations

$\begin{cases} x-y= 24, \\ x^2-2xy-y=24, \end{cases}$ what is the value of $$x+y$$?

Let $$a$$ and $$b$$ be the values of positive integers $$x$$ and $$y$$, respectively, that satisfy $x-y=5, x^2+y^2=73.$ What is the value of $$5a+8b?$$

If there exists exactly one solution $$(x, y)$$ that satisfies the simultaneous equations $x-14y=-1, x^2+xy+m=0,$ what is the value of $$\frac{1}{m}$$?

For the simultaneous equations $x+y = 2k+16, xy+x+y = k^2+5k+16$ to have real solutions, it must be the case that $$k \geq -\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

For what value of $$k$$ does there exist exactly one pair $$(x, y)$$ that satisfies the simultaneous equations $y-14x=k, x^2+y=5?$

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