Toán Lớp 7: Cho x, y, z khác $0$ thỏa mãn (x+y-z)/(z)=(y+z-x)/(x)=(z+x-y)/(y). Tính giá trị biểu thức P=(1+y/x) (1+z/y) (1+x/z)
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Toán Lớp 7: Cho x, y, z khác $0$ thỏa mãn (x+y-z)/(z)=(y+z-x)/(x)=(z+x-y)/(y). Tính giá trị biểu thức P=(1+y/x) (1+z/y) (1+x/z)
Comments ( 1 )
(x+y-z)/z=(y+z-x)/x=(z+x-y)/y=(x+y-z+y+z-x+z+x-y)/(x+y+z)=((x-x+x)+(y+y-y)+(z+z-z))/(x+y+z)=(x+y+z)/(x+y+z)=1
Do đó:
(x+y-z)/z=1=>x+y-z=z=>x+y=2z
(y+z-x)/x=1=>y+z-x=x=>y+z=2x
(z+x-y)/y=1=>z+x-y=y=>z+x=2y
P=(1+y/x)(1+z/y)(1+x/z)
P=(x/x+y/x)(y/y+z/y)(z/z+x/z)
P=(x+y)/x . (y+z)/y . (x+z)/z
P=(2z)/x . (2x)/y . (2y)/z
P=(2z.2x.2y)/(xyz)
P=(8xyz)/(xyz)
P=8
Vậy P=8