Question Number 74801 by behi83417@gmail.com last updated on 30/Nov/19
$$\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{3}} =\mathrm{23}}\\{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{32}}\end{cases}\:\:\:\:\:\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}\:. \\ $$
Commented by MJS last updated on 30/Nov/19
$$\mathrm{if}\:\mathrm{you}\:\mathrm{plot}\:\mathrm{these}\:\mathrm{you}\:\mathrm{see}\:\mathrm{there}'\mathrm{s}\:\mathrm{only}\:\mathrm{one} \\ $$$$\mathrm{real}\:\mathrm{solution} \\ $$$${x}\approx\mathrm{2}.\mathrm{96773} \\ $$$${y}\approx\mathrm{2}.\mathrm{42114} \\ $$
Commented by MJS last updated on 01/Dec/19
$$\mathrm{approximating}\:\mathrm{I}\:\mathrm{also}\:\mathrm{found}\:\mathrm{these} \\ $$$${x}\approx−\mathrm{1}.\mathrm{88268}\pm\mathrm{2}.\mathrm{70319i};\:{y}\approx−\mathrm{1}.\mathrm{83861}\pm\mathrm{2}.\mathrm{44516i} \\ $$$${x}\approx−\mathrm{1}.\mathrm{50880}\pm\mathrm{3}.\mathrm{04643i};\:{y}\approx−\mathrm{1}.\mathrm{29910}\mp\mathrm{2}.\mathrm{87425i} \\ $$$${x}\approx−\mathrm{1}.\mathrm{36613}\pm\mathrm{2}.\mathrm{49017i};\:{y}\approx\mathrm{3}.\mathrm{03262}\pm.\mathrm{247148i} \\ $$$${x}\approx\mathrm{3}.\mathrm{27375}\pm.\mathrm{139609i};\:{y}\approx−\mathrm{1}.\mathrm{10548}\pm\mathrm{2}.\mathrm{02898i} \\ $$$$…\mathrm{so}\:\mathrm{we}\:\mathrm{get}\:\mathrm{9}\:\mathrm{solutions} \\ $$
Answered by behi83417@gmail.com last updated on 01/Dec/19
$$\mathrm{dear}\:\mathrm{proph}.\:\mathrm{MJS}!\:\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}. \\ $$$$\mathrm{god}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Answered by ajfour last updated on 01/Dec/19
$$\:\left({x}^{\mathrm{2}} −\mathrm{23}\right)^{\mathrm{2}} =\left(\mathrm{32}−{x}^{\mathrm{3}} \right)^{\mathrm{3}} \\ $$$$\:\:{x}^{\mathrm{9}} −\mathrm{96}{x}^{\mathrm{6}} +{x}^{\mathrm{4}} +\mathrm{3072}{x}^{\mathrm{3}} −\mathrm{46}{x}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{23}^{\mathrm{2}} −\mathrm{32}^{\mathrm{3}} \:=\:\mathrm{0} \\ $$$${or}\:\:\:\:{x}^{\mathrm{2}} \left({x}−\mathrm{1}\right)+{y}^{\mathrm{2}} \left(\mathrm{1}−{y}\right)=\mathrm{9}\:={b}−{a} \\ $$$$\:\:\:\:\:\:\:\:\:{x}^{\mathrm{2}} \left({x}+\mathrm{1}\right)+{y}^{\mathrm{2}} \left(\mathrm{1}+{y}\right)=\mathrm{55}\:={a}+{b} \\ $$$$\Rightarrow\:\:{x}^{\mathrm{2}} =\frac{\left({b}−{a}\right)\left(\mathrm{1}+{y}\right)−\left({a}+{b}\right)\left(\mathrm{1}−{y}\right)}{\mathrm{2}\left({xy}−\mathrm{1}\right)} \\ $$$$\:\:\:\:\:\:{y}^{\mathrm{2}} =\frac{\left({a}+{b}\right)\left({x}−\mathrm{1}\right)−\left({b}−{a}\right)\left({x}+\mathrm{1}\right)}{\mathrm{2}\left({xy}−\mathrm{1}\right)} \\ $$$$\Rightarrow\:\:\frac{{x}^{\mathrm{2}} }{{y}^{\mathrm{2}} }=\frac{{by}−{a}}{{ax}−{b}}\:\:\:\:\:\& \\ $$$$\:\:\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \left({xy}−\mathrm{1}\right)=\left({by}−{a}\right)\left({ax}−{b}\right) \\ $$$${let}\:\:\:{x}=\frac{{b}}{{a}}{u}\:,\:\:{y}=\frac{{a}}{{b}}{v}\:\:\:\Rightarrow \\ $$$$\:\:\:{u}^{\mathrm{2}} {v}^{\mathrm{2}} \left({uv}−\mathrm{1}\right)={ab}\left({u}−\mathrm{1}\right)\left({v}−\mathrm{1}\right) \\ $$$$\:\:\:\:\frac{{b}^{\mathrm{4}} }{{a}^{\mathrm{4}} }=\frac{{a}}{{b}}\left(\frac{{v}−\mathrm{1}}{{u}−\mathrm{1}}\right) \\ $$$$\Rightarrow\:\:\frac{{v}−\mathrm{1}}{{u}−\mathrm{1}}=\frac{{b}^{\mathrm{5}} }{{a}^{\mathrm{5}} }\:=\:\frac{{k}}{\mathrm{1}} \\ $$$$\Rightarrow\:\:\frac{{u}+{v}−\mathrm{2}}{\:\sqrt{{uv}−\left({u}+{v}\right)+\mathrm{1}}}=\frac{{k}+\mathrm{1}}{\:\sqrt{{k}}} \\ $$$${let}\:\:{u}+{v}={s}\:,\:{uv}={m} \\ $$$$\Rightarrow\:\:{k}\left({s}−\mathrm{2}\right)^{\mathrm{2}} =\left({k}+\mathrm{1}\right)^{\mathrm{2}} \left\{\left({m}−\mathrm{1}\right)−\left({s}−\mathrm{2}\right)\right\} \\ $$$$\&\:\:\:{m}^{\mathrm{2}} \left({m}−\mathrm{1}\right)={ab}\left\{\left({m}−\mathrm{1}\right)−\left({s}−\mathrm{2}\right)\right\} \\ $$$$\Rightarrow\:{ab}\left(\mathrm{1}−\frac{{s}−\mathrm{2}}{{m}−\mathrm{1}}\right)={m}^{\mathrm{2}} \\ $$$$\:{k}\left({m}−\mathrm{1}\right)\left(\mathrm{1}−\frac{{m}^{\mathrm{2}} }{{ab}}\right)^{\mathrm{2}} =\frac{{m}^{\mathrm{2}} \left({k}+\mathrm{1}\right)^{\mathrm{2}} }{{ab}} \\ $$$$…… \\ $$$$…\:\left({matter}\:{of}\:{quintic}\:{polynomial}\right) \\ $$