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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.

Level 2

         

\[\begin{cases} { x }^{ 2 }+{ y }^{ 2 }=30 \\ x+y=10 \end{cases}\]

If the above equations are true simultaneously, then find the value of \(xy\).

\[ \large \begin{cases} { a(b+c)=32 } \\ { b(c+a) = 65 } \\ {c(a+b) = 77 } \end{cases}\]

Given that \(a , b,\) and \( c\) are positive real numbers that satisfy the system of equations above, find the value of \(abc\).

Given that \[{ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }+{ d }^{ 2 }=1\] And also that \( ac+bd=0\) What is the value of \(ab+cd\)?

For the positive numbers \(d, o, v, e\): \[d^2 + do = 250 \\ o^2 + do = 150\] \[v^2 + ve = 30 \\ e^2 + ve = 34\] Find the value of \(d + o + v + e\).

You think you rock at linear equation? Gimme the solution NOT using hit and trial method! Solve for the values of "x" and "y" and enter \(x+y\)

Note: These are not linear equations.

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