已知1≤x+y≤5,-1≤x-y≤3,求2x-4y的范围

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已知1≤x+y≤5,-1≤x-y≤3,求2x-4y的范围
已知1≤x+y≤5,-1≤x-y≤3,求2x-4y的范围

已知1≤x+y≤5,-1≤x-y≤3,求2x-4y的范围
设2x-4y=m(x+y)+n(x-y)=mx+my+nx-ny=(m+n)x+(m-n)y
所以m+n=2,m-n=-4,所以m=-1,n=3,
所以2x-4y=-(x+y)+3(x-y),
由1≤x+y≤5,得-5《-(x+y)《-1,
由-1≤x-y≤3,得-3《3(x-y)《9,两式相加,得
-8《-(x+y)+3(x-y)《8,
即-8《2x-4y≤8

两式加得:0<=2x<=8
,-1≤x-y≤3, (1)
-3<=y-x<=1 且1≤x+y≤5
上两式加得:-2<=2y<=6
-6<=-2y<=2 (2)
(1)(2)加:t
-5<=2x-2y<=5

(-4,12)