For x ,y ε Z^+ such that 7x+9y=405. Find max value of x−y.


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Question Number 175799 by cortano1 last updated on 07/Sep/22

$For x, y ε Z^{+} such that$ $7 x + 9 y = 405 . Find \max value$ $of x - y .$
	Answered by mr W last updated on 07/Sep/22

	$7 x + 9 y = 405$ $\Rightarrow x = 9 k + 54 > 0 \Rightarrow k ⩾ - 5$ $\Rightarrow y = - 7 k + 3 > 0 \Rightarrow k ⩽ 0$ $s = x - y = 9 k + 54 + 7 k - 3 = 16 k + 51$ $s_{m i n} = - 16 \times 5 + 51 = - 29$ $s_{m a x} = 16 \times 0 + 51 = 51$
	Commented by cortano1 last updated on 07/Sep/22

	$nice$
	Answered by blackmamba last updated on 07/Sep/22

	$l e t h = x - y a n d x = h + y$ $g i v e n c o n d i t i o n 7 x + 9 y = 405$ $\Rightarrow 7 (h + y) + 9 y = 405$ $\Rightarrow h = \frac{405 - 16 y}{7} = \frac{406 - 1 - 14 y - 2 y}{7}$ $\Rightarrow h = 58 - 2 y - \frac{2 y + 1}{7}$ $h m a x w h e n y m i n; y = 3$ $s o h_{m a x} = 58 - 6 - 1 = 51$
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