Tìm x,y,z thoả mãn
|7x-5y|+ |2z-3x|+|xy+yz+zx-2000|=0
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Nhận xét:{(|7x-5y|>=0AAx;y),(|2z-3x|>=0AAx;z),(|xy+yz+zx-2000|>=0AAx;y;z):}=>|7x-5y|+|2z-3x|+|xy+yz+zx-2000|>=0AAx;y;zmà theo đề |7x-5y|+|2z-3x|+|xy+yz+zx-2000|=0nên dấu bằng xảy ra <=>{(7x=5y(1)),(2z=3x(2)),(xy+yz+zx=2000(3)):}Xét (1) và (2) ta có:@7x=5y=>x/5=y/7=>x/10=y/14@2z=3x=>x/2=z/3=>x/10=z/15Do đó: x/10=y/14=z/15=>x;y;z cùng dấu.Đặt x/10=y/14=z/15=k=>{(x=10k),(y=14k),(z=15k):}, thế vào (3) ta được:xy+yz+zx=2000<=>10k.14k+14k.15k+15k.10k=2000<=>140k^2+210k^2+150k^2=2000<=>500k^2=2000<=>k^2=4<=>k=+-2Khi đó: (x;y;z)=(+-20);+-28;+-30) với x;y;z cùng dấu.
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Vì |7x – 5y| \ge 0 AAx,y|2z – 3x| \ge 0 AAz,x|xy + yz + zx – 2000| \ge 0 AAx,y,z=> |7x – 5y| + |2z – 3x| + |xy + yz + zx – 2000| \ge 0Dấu “=” xảy ra <=> {(|7x – 5y| = 0),(|2z – 3x| = 0),(|xy + yz + zx – 2000| = 0):}<=> {(7x – 5y = 0),(2z – 3x = 0),(xy + yz + zx – 2000 = 0):}<=> {(7x = 5y),(2z = 3x),(xy+ yz + zx = 2000):}<=> {(x/5 = y/7),(z/3 = x/2),(xy + yz + zx = 2000):}<=> {(x/(10) = y/(14)),(z/(15) = x/(10)),(xy + yz + zx = 2000):}<=> {(x/(10) = y/(14) = z/(15)),(xy + yz + zx = 2000):}Đặt x/(10) = y/(14) = z/(15) = k=> {(x =10k),(y = 14k),(z = 15k):}Thay x = 10k,y = 14k,z = 15k vào xy + yz + zx = 2000 ta đc:10k.14k + 14k.15k + 15k.10k = 2000<=> 140k^2 + 210k^2 + 150k^2 = 2000<=> 500k^2 = 2000<=> k^2 = 4<=> k = +-2+) k = 2=> {(x = 10.2),(y = 14.2),(z = 15.2):}=> {(x = 20),(y = 28),(z = 30):}+) k = -2=> {(x = 10.(-2)),(y = 14.(-2)),(z = 15.(-2)):}=> {(x = -20),(y = -28),(z = -30):}Vậy (x,y,z) \in {(20,28,30),(-20,-28,-30)}$#duong612009$