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Quadratic Simultaneous Equations With Mixed and Quadratic Terms in Both Equations

Simultaneous Equations with square and mixed
\[xy\]
terms in both equations can be quite tricky to solve.
\[x^2+xy=30\]
(1)
\[y^2+xy=20\]
(1)
Adding these gives
\[x^2-xy+y^2+xy=50 \rightarrow x^2+2xy+y^2=50\]

We can factorise as
\[(x+y)^2=50 \rightarrow x+y=\pm \sqrt{50}=\pm5 \sqrt{2}\]
(3)
(1)-(2) gives
\[x^2-y^2=10 \rightarrow (x+y)(x-y)=10\]
(4)
(4) divided by (3) gives
\[x-y=\pm \sqrt{2}\]
(5)
Now we have ordinary simultaneous equations.
\[x+y=\pm5 \sqrt{2}\]

\[x-y=\pm\sqrt{2}\]

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