Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 42296 by maxmathsup by imad last updated on 22/Aug/18

solve in Z^2 2x+3y =7

$${solve}\:{in}\:{Z}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{2}{x}+\mathrm{3}{y}\:=\mathrm{7} \\ $$

Commented by math khazana by abdo last updated on 22/Aug/18

we have 2(−1)+3(3) =7 so (−1,3)is a special solution (e) ⇒2x+3y−2(−1)−3(3)=0 ⇒ 2(x+1) +3(y+3) =0 ⇒2(x+1) =−3(y+3) ⇒ 3 divide 2(x+1) but D(2,3)=1 ⇒3divide x+1 ⇒ x+1=3k ⇒x= 3k−1 ⇒3y =7−2x =7−2(3k−1)=7−6k+2=9−6k ⇒ y =3−2k finally x =3k−1 and y=−2k+3 with k∈Z .

$${we}\:{have}\:\mathrm{2}\left(−\mathrm{1}\right)+\mathrm{3}\left(\mathrm{3}\right)\:=\mathrm{7}\:{so}\:\left(−\mathrm{1},\mathrm{3}\right){is}\:{a}\:{special} \\ $$$${solution}\:\left({e}\right)\:\Rightarrow\mathrm{2}{x}+\mathrm{3}{y}−\mathrm{2}\left(−\mathrm{1}\right)−\mathrm{3}\left(\mathrm{3}\right)=\mathrm{0}\:\Rightarrow \\ $$$$\mathrm{2}\left({x}+\mathrm{1}\right)\:+\mathrm{3}\left({y}+\mathrm{3}\right)\:=\mathrm{0}\:\Rightarrow\mathrm{2}\left({x}+\mathrm{1}\right)\:=−\mathrm{3}\left({y}+\mathrm{3}\right)\:\Rightarrow \\ $$$$\mathrm{3}\:{divide}\:\mathrm{2}\left({x}+\mathrm{1}\right)\:{but}\:{D}\left(\mathrm{2},\mathrm{3}\right)=\mathrm{1}\:\Rightarrow\mathrm{3}{divide}\:{x}+\mathrm{1}\:\Rightarrow \\ $$$${x}+\mathrm{1}=\mathrm{3}{k}\:\Rightarrow{x}=\:\mathrm{3}{k}−\mathrm{1}\:\Rightarrow\mathrm{3}{y}\:=\mathrm{7}−\mathrm{2}{x} \\ $$$$=\mathrm{7}−\mathrm{2}\left(\mathrm{3}{k}−\mathrm{1}\right)=\mathrm{7}−\mathrm{6}{k}+\mathrm{2}=\mathrm{9}−\mathrm{6}{k}\:\Rightarrow \\ $$$${y}\:=\mathrm{3}−\mathrm{2}{k}\:{finally}\:{x}\:=\mathrm{3}{k}−\mathrm{1}\:{and}\:{y}=−\mathrm{2}{k}+\mathrm{3}\:{with} \\ $$$${k}\in{Z}\:. \\ $$

Answered by ajfour last updated on 22/Aug/18

2(x+y)+y=7 let x+y=n ⇒ 2n+y = 7 ⇒ y=7−2n x=3n−7 (n∈ Z ).

$$\mathrm{2}\left({x}+{y}\right)+{y}=\mathrm{7} \\ $$$${let}\:\:{x}+{y}={n} \\ $$$$\Rightarrow\:\:\mathrm{2}{n}+{y}\:=\:\mathrm{7} \\ $$$$\Rightarrow\:{y}=\mathrm{7}−\mathrm{2}{n} \\ $$$$\:\:\:\:\:\:{x}=\mathrm{3}{n}−\mathrm{7}\:\:\:\:\left({n}\in\:\mathbb{Z}\:\right). \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com