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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.

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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