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Question Number 78755 by TawaTawa last updated on 20/Jan/20

Solve the equation:      xy + 5x + 5y  =  − 25      ... (i)      yz + 3y + 5z  =  − 15      ... (ii)      xz + 5z + 3x  =  − 15      ... (iii)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\mathrm{xy}\:+\:\mathrm{5x}\:+\:\mathrm{5y}\:\:=\:\:−\:\mathrm{25}\:\:\:\:\:\:...\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\mathrm{yz}\:+\:\mathrm{3y}\:+\:\mathrm{5z}\:\:=\:\:−\:\mathrm{15}\:\:\:\:\:\:...\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\mathrm{xz}\:+\:\mathrm{5z}\:+\:\mathrm{3x}\:\:=\:\:−\:\mathrm{15}\:\:\:\:\:\:...\:\left(\mathrm{iii}\right) \\ $$

Commented by Tony Lin last updated on 20/Jan/20

xy+5x+5y+25=0  (x+5)(y+5)=0...(i)  yz+3y+5z+15=0  (y+5)(z+3)=0...(ii)  xz+5z+3x+15=0  (x+5)(z+3)=0...(iii)  ⇒ { ((x=−5 ∨ y=−5)),((y=−5 ∨ z=−3)),((x=−5 ∨ z=−3)) :}  ⇒(x=−5∧y=−5)  ∨(y=−5∧z=−3)  ∨(x=−5∧z=−3)

$${xy}+\mathrm{5}{x}+\mathrm{5}{y}+\mathrm{25}=\mathrm{0} \\ $$$$\left({x}+\mathrm{5}\right)\left({y}+\mathrm{5}\right)=\mathrm{0}...\left({i}\right) \\ $$$${yz}+\mathrm{3}{y}+\mathrm{5}{z}+\mathrm{15}=\mathrm{0} \\ $$$$\left({y}+\mathrm{5}\right)\left({z}+\mathrm{3}\right)=\mathrm{0}...\left({ii}\right) \\ $$$${xz}+\mathrm{5}{z}+\mathrm{3}{x}+\mathrm{15}=\mathrm{0} \\ $$$$\left({x}+\mathrm{5}\right)\left({z}+\mathrm{3}\right)=\mathrm{0}...\left({iii}\right) \\ $$$$\Rightarrow\begin{cases}{{x}=−\mathrm{5}\:\vee\:{y}=−\mathrm{5}}\\{{y}=−\mathrm{5}\:\vee\:{z}=−\mathrm{3}}\\{{x}=−\mathrm{5}\:\vee\:{z}=−\mathrm{3}}\end{cases} \\ $$$$\Rightarrow\left({x}=−\mathrm{5}\wedge{y}=−\mathrm{5}\right) \\ $$$$\vee\left({y}=−\mathrm{5}\wedge{z}=−\mathrm{3}\right) \\ $$$$\vee\left({x}=−\mathrm{5}\wedge{z}=−\mathrm{3}\right) \\ $$

Commented by TawaTawa last updated on 20/Jan/20

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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