<s>[INST] Olympiad Math Problem About Simultaneous Equations Given the question:

*

*$x + y = 1$

*$x^2 + y^2 = 2$

*$x^5 + y^5 = ?$
After a bunch of manipulation of the above equations we find the solution to be 19/4. Yet could the above be solved by simultaneous equations?
From $1)$ we can conclude that $x = 1 - y$.
Substituting this into $2)$ we get a quadratic $2y^2 - 2y - 1 = 0$.
However, when we solve for $y$ we get two possible solutions. Hence how do we proceed from here? Why doesn't simultaneous equations work to solve the above problem?
 [/INST] $$1=(x+y)^2= x^2+2xy+y^2 = 2+2xy$$$$ \implies xy = -1/2$$
$$4=(x^2+y^2)^2 =x^4+2x^2y^2 +y^4$$$$\implies x^4+y^4= 7/2$$
and $$2=(x+y)(x^2+y^2) = x^3+xy(x+y)+y^3$$ $$\implies x^3+y^3 = 5/2
$$
So
$$7/2=(x+y)(x^4+y^4) = x^5+y^5 + xy(x^3+y^3)$$ $$\implies x^5+y^5 =19/4$$
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