Quadratic Simultaneous Equations With Mixed and Quadratic Terms in Both Equations

Simultaneous Equations with square and mixed
terms in both equations can be quite tricky to solve.
Adding these gives
\[x^2-xy+y^2+xy=50 \rightarrow x^2+2xy+y^2=50\]

We can factorise as
\[(x+y)^2=50 \rightarrow x+y=\pm \sqrt{50}=\pm5 \sqrt{2}\]
(1)-(2) gives
\[x^2-y^2=10 \rightarrow (x+y)(x-y)=10\]
(4) divided by (3) gives
\[x-y=\pm \sqrt{2}\]
Now we have ordinary simultaneous equations.
\[x+y=\pm5 \sqrt{2}\]


Adding these gives
\[2x=\pm 6 \sqrt{2} \rightarrow x = \pm 3 \sqrt{5}\]
Subtracting these gives
\[2y=\pm 4 \sqrt{2} \rightarrow y = \pm 2 \sqrt{5}\]

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