Question Number 84188 by jagoll last updated on 10/Mar/20
$$\mathrm{if}\:\mathrm{2x}+\mathrm{3y}\:=\:\mathrm{2020}? \\ $$$$\mathrm{find}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{3x}+\mathrm{2y}\:\mathrm{for}\:\mathrm{x}\:\mathrm{and}\:\mathrm{natural} \\ $$$$\mathrm{number} \\ $$
Commented by jagoll last updated on 10/Mar/20
$$\mathrm{this}\:\mathrm{diopthantine}\:\mathrm{equation}\:? \\ $$
Commented by jagoll last updated on 10/Mar/20
$$\mathrm{yes}.\:\mathrm{my}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{correct}.\:\mathrm{but}\:\mathrm{my}\:\mathrm{way} \\ $$$$\mathrm{is}\:\mathrm{different}\:\mathrm{with}\:\mathrm{you}\:\mathrm{way}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 10/Mar/20
$${x}=−\mathrm{3}{k}+\mathrm{1007}\geqslant\mathrm{1} \\ $$$${y}=\mathrm{2}{k}+\mathrm{2}\geqslant\mathrm{1} \\ $$$$\Rightarrow\mathrm{0}\leqslant{k}\leqslant\mathrm{335} \\ $$$$ \\ $$$$\mathrm{3}{x}+\mathrm{2}{y}=\mathrm{3}\left(−\mathrm{3}{k}+\mathrm{1007}\right)+\mathrm{2}\left(\mathrm{2}{k}+\mathrm{2}\right) \\ $$$$=\mathrm{3025}−\mathrm{5}{k}\:\leqslant\mathrm{3025} \\ $$$$=\mathrm{3025}−\mathrm{5}{k}\:\geqslant\mathrm{1350} \\ $$$$ \\ $$$${i}.{e}.\:{maximum}\:{of}\:\mathrm{3}{x}+\mathrm{2}{y}\:{is}\:\mathrm{3025}. \\ $$$${i}.{e}.\:{minimum}\:{of}\:\mathrm{3}{x}+\mathrm{2}{y}\:{is}\:\mathrm{1350}. \\ $$
Commented by mr W last updated on 11/Mar/20
$${yes}.\:{there}\:{are}\:{many}\:{ways}\:{to}\:{solve}. \\ $$$${i}\:{not}\:{only}\:{wanted}\:{to}\:{know}\:{the}\: \\ $$$${minimum}\:{and}\:{maximum},\:{but}\:{also} \\ $$$${wanted}\:{to}\:{know}\:{the}\:{general}\:{solution} \\ $$$${of}\:{the}\:{equation}\:{and}\:{how}\:{many}\:{solutions} \\ $$$${it}\:{has}.\:{therefore}\:{i}\:{selected}\:{this}\:{way}. \\ $$