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Question Number 66756 by John Kaloki Musau last updated on 19/Aug/19
solve the equations,  x+y=17  xy−5x=32
$${solve}\:{the}\:{equations}, \\ $$$${x}+{y}=\mathrm{17} \\ $$$${xy}−\mathrm{5}{x}=\mathrm{32} \\ $$
Answered by $@ty@m123 last updated on 19/Aug/19
x(17−x)−5x=32  17x−x^2 −5x=32  x^2 −12x+32=0  x=((12±(√(144−128)))/2)  x=((12±4)/2) = 8,4  ⇒y=9,13
$${x}\left(\mathrm{17}−{x}\right)−\mathrm{5}{x}=\mathrm{32} \\ $$$$\mathrm{17}{x}−{x}^{\mathrm{2}} −\mathrm{5}{x}=\mathrm{32} \\ $$$${x}^{\mathrm{2}} −\mathrm{12}{x}+\mathrm{32}=\mathrm{0} \\ $$$${x}=\frac{\mathrm{12}\pm\sqrt{\mathrm{144}−\mathrm{128}}}{\mathrm{2}} \\ $$$${x}=\frac{\mathrm{12}\pm\mathrm{4}}{\mathrm{2}}\:=\:\mathrm{8},\mathrm{4} \\ $$$$\Rightarrow{y}=\mathrm{9},\mathrm{13} \\ $$
Answered by John Kaloki Musau last updated on 20/Aug/19
x=17−y  y(17−y)−5(17−y)=32  17y−y^2 −85+5y=32  22y−y^2 −117=0  y^2 −13y−9y+117=0  y(y−13)−9(y−13)=0  (y−9)(y−13)=0  y=9 and 13
$$\boldsymbol{{x}}=\mathrm{17}−{y} \\ $$$$\boldsymbol{{y}}\left(\mathrm{17}−\boldsymbol{{y}}\right)−\mathrm{5}\left(\mathrm{17}−\boldsymbol{{y}}\right)=\mathrm{32} \\ $$$$\mathrm{17}\boldsymbol{{y}}−\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{85}+\mathrm{5}\boldsymbol{{y}}=\mathrm{32} \\ $$$$\mathrm{22}\boldsymbol{{y}}−\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{117}=\mathrm{0} \\ $$$$\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{13}\boldsymbol{{y}}−\mathrm{9}\boldsymbol{{y}}+\mathrm{117}=\mathrm{0} \\ $$$$\boldsymbol{{y}}\left(\boldsymbol{{y}}−\mathrm{13}\right)−\mathrm{9}\left(\boldsymbol{{y}}−\mathrm{13}\right)=\mathrm{0} \\ $$$$\left(\boldsymbol{{y}}−\mathrm{9}\right)\left(\boldsymbol{{y}}−\mathrm{13}\right)=\mathrm{0} \\ $$$$\boldsymbol{{y}}=\mathrm{9}\:\boldsymbol{{and}}\:\mathrm{13} \\ $$

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