Please help. Find all the general solution of 6x + 8y + 5z = 101 . I got: x = − 48 + 45m + 4n y = 48 + 45m − 3n z = 1 − 2m Please help. Am confused.


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Question Number 46157 by Tawa1 last updated on 21/Oct/18

$Please help .$ $Find all the general solution of 6 x + 8 y + 5 z = 101 .$ $I got : x = - 48 + 45 m + 4 n$ $y = 48 + 45 m - 3 n$ $z = 1 - 2 m$ $Please help . Am confused .$
	Answered by MrW3 last updated on 22/Oct/18

	$6 x + 8 y + 5 z = 101 w i t h x, y, z \in Z$ $s i n c e g c d (6, 8, 5) = 1 w h i c h d i v i d e s 101,$ $\Rightarrow s o l u t i o n e x i s t s!$ $l e t 3 x + 4 y = t . . . (i)$ $\Rightarrow 2 t + 5 z = 101 . . . (i i)$ $(i i) h a s a p a r t i c u l a r s o l u t i o n$ $t_{1} = 48, z_{1} = 1$ $s o i t s g e n e r a l s o l u t i o n i s$ $t = 48 + 5 m$ $z = 1 - 2 m$ $3 x + 4 y = 1 h a s a p a r t i c u l a r s o l u t i o n$ $x_{1} = - 1, y_{1} = 1$ $s o i t s g e n e r a l s o l u t i o n i s$ $x = - 1 + 4 k$ $y = 1 - 3 k$ $t h e g e n e r a l s o l u t i o n f o r (i) i s t h e n$ $x = (- 1 + 4 k) (48 + 5 m) = - 48 - 5 m + 4 k (48 + 5 m)$ $y = (1 - 3 k) (48 + 5 m) = 48 + 5 m - 3 k (48 + 5 m)$ $l e t n = k (48 + 5 m)$ $w e g e t t h e g e n e r a l s o l u t i o n$ $x = - 48 - 5 m + 4 n$ $y = 48 + 5 m - 3 n$ $z = 1 - 2 m$ $c h e c k :$ $6 (- 48 - 5 m + 4 n) + 8 (48 + 5 m - 3 n) + 5 (1 - 2 m)$ $= - 288 - 30 m + 24 n + 384 + 40 m - 24 n + 5 - 10 m$ $= 101 \Rightarrow c o r r e c t!$
	Commented by Tawa1 last updated on 21/Oct/18

	$Ohh, i got it but slight mistake . i followed your method earlier .$
	Commented by Tawa1 last updated on 21/Oct/18

	$Sir, i will always set t = 3 x + 4 y, later 3 x + 4 y = 1 ? ? .$ $Sir why not 3 x + 4 y = 101 ?$
	Commented by Tawa1 last updated on 21/Oct/18

	$And why is the general solution for (i) is x \times u$ $i think those are the areas i need to grasp$
	Commented by MrW3 last updated on 23/Oct/18

	$i n t h e f i r s t s t e p y o u f i n d t h e s o l u t i o n$ $f o r 2 t + 5 z = 101 w h i c h i s t = 48 + 5 m .$ $i n n e x t s t e p y o u s h o u l d f i n d t h e$ $s o l u t i o n f o r 3 x + 4 y = 48 + 5 m . b u t t h i s$ $i s n o t e a s y . s o y o u f i n d a t f i r s t t h e$ $s o l u t i o n f o r 3 x + 4 y = 1 w h i c h i s x = - 1 + 4 k .$ $t h e s o l u t i o n f o r 3 x + 4 y = 48 + 5 m i s t h e n$ $x = (- 1 + 4 k) (48 + 5 m) = - 48 - 5 m + 4 k (48 + 5 m)$ $= - 48 - 5 m + 4 n .$
	Commented by Tawa1 last updated on 21/Oct/18

	$Wow, God bless you sir . i understand now$
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