if-3x-5y-1-use-Bezout-s-identity-to-find-the-value-of-x-and-y-

Question Number 64086 by Rio Michael last updated on 12/Jul/19

i f 3 x + 5 y = 1

u s e {B e z o u t}^{'} s i d e n t i t y t o f i n d t h e v a l u e o f x a n d y

Commented by turbo msup by abdo last updated on 13/Jul/19

l e t c o n s i d e r c o n g r u e n c e [3] (Z / 3 Z)

(e) \Rightarrow \bar{3} \bar{x} + \bar{5} \bar{y} = \bar{1} \Rightarrow

0 + \bar{2} \bar{y} = \bar{1} = - \bar{2} \Rightarrow \bar{y} = - \bar{1} \Rightarrow

y = - 1 + 3 k k i n t e g r \Rightarrow

x = \frac{1 - 5 y}{3} = \frac{1 + 5 - 15 k}{3} = 2 - 5 k \Rightarrow

t h e s o l u t i o n a r e (2 - 5 k, - 1 + 3 k)

k f r o m Z .

Commented by turbo msup by abdo last updated on 13/Jul/19

a n o t h e r w a y b y p a r t i c u l a r s o l u t i o n

(2, - 1) i s a o s r t i c u l a r s o l u t i o n

w e h a v e 3 x + 5 y = 1 a n d

3 \times 2 + 5 \times (- 1) = 1 \Rightarrow

3 (x - 2) + 5 (y + 1) = 0 \Rightarrow

3 (x - 2) = - 5 (y + 1) \Rightarrow

5 d i v i d e 3 (x - 2) b u t Δ (3, 5) = 1 \Rightarrow

5 d i v i d e x - 2 \Rightarrow x = 5 k + 2

w e h a v e y = \frac{1 - 3 x}{5} = \frac{1 - 15 k - 6}{5}

= - 1 - 3 k s o (5 k + 2, - 1 - 3 k) a r e

s o l u t i o n f o r t h i s e q u a t i o n .

Commented by Rio Michael last updated on 13/Jul/19

i t h o u g h t w e a r e t o g e t c o n s t a n t s a s a n s w e r s .

w e u s e g c d (3, 5) = 1 r i g h t ? s i n c e t h e y a r e r e l a t i v e y p r i m e

Commented by mr W last updated on 13/Jul/19

s e e a l s o Q 46592

Commented by Rio Michael last updated on 13/Jul/19

y e s i s a w t h a t, b u t m u s t w e a l w a y s p u t i t i n a g e n e r a l s o l u t i o n

s i n c e g c d (3, 5) = 1

t h e n w e r e v e r s e r i g h t

x = - 2 a n d y = 1