Find the general solution of : 311x − 112y = 73


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Question Number 44704 by Tawa1 last updated on 03/Oct/18

$Find the general solution of : 311 x - 112 y = 73$
	Answered by Joel578 last updated on 04/Oct/18

	$311 x - 112 y = 73$ $311 x = 73 + 112 y$ $x = \frac{73}{311} + \frac{112}{311} y$ $Let y = t$ $\to x = \frac{73}{311} + \frac{112}{311} t$ $General solution$ $(x, y) = (\frac{73}{311} + \frac{112}{311} t, t)$ $Where t \in R$
	Commented by Tawa1 last updated on 03/Oct/18

	$How sir$
	Commented by MrW3 last updated on 04/Oct/18

	$I t h i n k y o u m e a n a g e n e r a l i n t e g e r$ $s o l u t i o n . I t i s t h e n$ $x = 15 + 112 n$ $y = 41 + 311 n$ $w i t h n = a n y i n t e g e r$ $p l e a s e h a v e a l o o k a t Q 19198 .$
	Commented by Tawa1 last updated on 04/Oct/18

	$God bless you sir . i will go through your explanation and get back sir$
	Commented by Tawa1 last updated on 04/Oct/18

	$Please sir . brush the solution . i will study it sir$
	Answered by MrW3 last updated on 05/Oct/18

	$H o w t o f i n d t h e g e n e r a l s o l u t i o n o f$ $311 x - 112 y = 73 w i t h x, y \in Z$ $S t e p 1 : f i n d t h e g c d o f 311 a n d 112 .$ $311 = 2 \times 112 + 87$ $112 = 1 \times 87 + 25$ $87 = 3 \times 25 + 12$ $25 = 2 \times 12 + 1 \leftarrow g c d f r o m 311 a n 112$ $12 = 12 \times 1 + 0$ $\Rightarrow g c d (311, 112) = 1 a n d t h i s d i v i d e s 73$ $\Rightarrow i n f i n i t e s o l u t i o n s e x i s t .$ $S t e p 2 : f i n d a p a r t i c u l a r s o l u t i o n .$ $73 = 3 \times 25 - 2 \times 1$ $= 3 \times 25 - 2 \times (25 - 2 \times 12)$ $= 25 + 4 \times 12$ $= 25 + 4 \times (87 - 3 \times 25)$ $= 4 \times 87 - 11 \times 25$ $= 4 \times 87 - 11 \times (112 - 1 \times 87)$ $= 15 \times 87 - 11 \times 112$ $= 15 \times (311 - 2 \times 112) - 11 \times 112$ $= 15 \times 311 - 41 \times 112$ $i . e . x = 15 a n d y = 41 i s a s o l u t i o n,$ $s i n c e 311 \times 15 - 112 \times 41 = 73$ $S t e p 3 : f i n d t h e g e n e r a l s o l u t i o n .$ $x = 15 + 112 n$ $y = 41 + 311 n w i t h n \in Z$ $T h i s c a n b e e a s i l y c h e c k e d :$ $311 \times (15 + 112 n) - 112 \times (41 + 311 n)$ $= (311 \times 15 - 112 \times 41) + (311 \times 112 n - 112 \times 311 n)$ $= 73 + 0$ $= 73$
	Commented by Tawa1 last updated on 04/Oct/18

	$Yes . God bless you sir . Thanks for your help sir$
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