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Question Number 42291 by Joel578 last updated on 22/Aug/18

y = ((1080 − 19x)/(49))        (x,y ∈ Z)  Find (x,y)

$${y}\:=\:\frac{\mathrm{1080}\:−\:\mathrm{19}{x}}{\mathrm{49}}\:\:\:\:\:\:\:\:\left({x},{y}\:\in\:\mathbb{Z}\right) \\ $$$$\mathrm{Find}\:\left({x},{y}\right) \\ $$

Answered by ajfour last updated on 22/Aug/18

19x+49y=1080  19(x+y)+30y=30×36  x+y = 30n     (n∈Z )  ⇒19n +y = 36  ⇒   n=1,  y=17 , x=13 .  also  if  n=2 , y=−2,  x=62  if  n=3,  y=−21 ,  x=111  if n=−1  , y=55 , x= −85  and  so on..

$$\mathrm{19}{x}+\mathrm{49}{y}=\mathrm{1080} \\ $$$$\mathrm{19}\left({x}+{y}\right)+\mathrm{30}{y}=\mathrm{30}×\mathrm{36} \\ $$$${x}+{y}\:=\:\mathrm{30}{n}\:\:\:\:\:\left({n}\in\mathbb{Z}\:\right) \\ $$$$\Rightarrow\mathrm{19}{n}\:+{y}\:=\:\mathrm{36} \\ $$$$\Rightarrow\:\:\:{n}=\mathrm{1},\:\:{y}=\mathrm{17}\:,\:{x}=\mathrm{13}\:. \\ $$$${also}\:\:{if}\:\:{n}=\mathrm{2}\:,\:{y}=−\mathrm{2},\:\:{x}=\mathrm{62} \\ $$$${if}\:\:{n}=\mathrm{3},\:\:{y}=−\mathrm{21}\:,\:\:{x}=\mathrm{111} \\ $$$${if}\:{n}=−\mathrm{1}\:\:,\:{y}=\mathrm{55}\:,\:{x}=\:−\mathrm{85} \\ $$$${and}\:\:{so}\:{on}.. \\ $$

Commented by Joel578 last updated on 22/Aug/18

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

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