Solve the simultaneous equations involving x, y, and their squares

[chwala]

Homework Statement

See attached

Relevant Equations

simultaneous equations

In my approach,

##y=6-x##

##x^2(6-x)^2+2x(6-x)-35=0##

...

##x^4-12x^3+34x^2+12x-35=0##

using factor theorem, i noted that ##f(1)=0## therefore ##(x-1)## is a factor.

on long division we also have other factors:##(x+1), (x-5)## and ##(x-7)## from these we can find ##y## values by using ##y=6-x##. This may be the only approach ....quite easy this one.

 

[Steve4Physics]

Let ##w = xy##, then ##x^2y^2 + 2xy - 35 = 0## becomes ##w^2 +2w - 35 = 0##.

##(w-5)(w+7) = 0##. I.e ##xy = 5## or ##xy=-7##. And take it from there.

 

[Mark44]

Similar to @Steve4Physics's approach, but factoring the quadratic in x and y first:
##x^2y^2 + 2xy - 35 = 0 \Rightarrow (xy + 7)(xy - 5) = 0##
##\Rightarrow xy = -7 \text{ or } xy = 5##

Solve ##x + y = 6## for y (or x) and substitute this expression into the equations above.