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Question Number 176 by 123456 last updated on 25/Jan/15

find the integer solution of 3x+4y=5

$$\mathrm{find}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{3x}+\mathrm{4y}=\mathrm{5} \\ $$

Answered by prakash jain last updated on 14/Dec/14

gcd(3,4)=1 and 1 divides 5 so it is solvable. Particular solution x=3, y=−1 General solution x=4k+3, y=−3k−1 where k is Integer For k=0, x=3, y=−1 k=1, x=7, y=−4 k=2, x=11, y=−7 k=−1, x=−1, y=2

$$\mathrm{gcd}\left(\mathrm{3},\mathrm{4}\right)=\mathrm{1}\:\mathrm{and}\:\mathrm{1}\:\mathrm{divides}\:\mathrm{5}\:\mathrm{so}\:\mathrm{it}\:\mathrm{is}\:\mathrm{solvable}. \\ $$$$\mathrm{Particular}\:\mathrm{solution}\:{x}=\mathrm{3},\:{y}=−\mathrm{1} \\ $$$$\mathrm{General}\:\mathrm{solution} \\ $$$${x}=\mathrm{4}{k}+\mathrm{3},\:{y}=−\mathrm{3}{k}−\mathrm{1}\:\:\:\mathrm{where}\:{k}\:\mathrm{is}\:\mathrm{Integer} \\ $$$$\mathrm{For}\:{k}=\mathrm{0},\:\:{x}=\mathrm{3},\:{y}=−\mathrm{1} \\ $$$${k}=\mathrm{1},\:{x}=\mathrm{7},\:{y}=−\mathrm{4} \\ $$$${k}=\mathrm{2},\:{x}=\mathrm{11},\:{y}=−\mathrm{7} \\ $$$${k}=−\mathrm{1},\:{x}=−\mathrm{1},\:{y}=\mathrm{2} \\ $$

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