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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.

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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