x-y-z-1-42x-44y-30z-42-x-y-z-1-0-0-yes-but-solution-

Question Number 209436 by hardmath last updated on 10/Jul/24

{\begin{cases} x + y + z = 1 \\ 42 x + 44 y + 30 z = 42 \end{cases}

(x, y, z) = (1, 0, 0) yes, but solution \dots

Commented by mr W last updated on 10/Jul/24

y o u m e a n i n t e g e r s o l u t i o n s .

Commented by hardmath last updated on 10/Jul/24

yes dear professor

Answered by mr W last updated on 10/Jul/24

z = 1 - x - y

42 x + 44 y + 30 z = 42

21 x + 22 y + 15 z = 21

21 x + 22 y + 15 (1 - x - y) = 21

\Rightarrow 7 y = 6 (1 - x)

\Rightarrow 1 - x = 7 k \Rightarrow x = 1 - 7 k

\Rightarrow y = 6 k

\Rightarrow z = 1 - 1 + 7 k - 6 k = k

g e n e r a l s o l u t i o n w i t h k \in Z :

x = 1 - 7 k

y = 6 k

z = k

Commented by hardmath last updated on 13/Jul/24

t hank y ou d ear p rofessor