$\quad \begin{cases}-2mx+y=4\ (1)\\2x+my=2\ (2)\end{cases}$
Từ `(1)=>y=2mx+4` thay vào `(2)`
`(2)<=>2x+m(2mx+4)=2`
`<=>2x+2m^2x+4m=2`
`<=>2(1+m^2)x=2(1-2m)`
`<=>x={1-2m}/{1+m^2}`
`\qquad y=2mx+4`
`<=>y=2m. {1-2m}/{1+m^2} +4`
`<=>y={2m(1-2m)+4(1+m^2)}/{1+m^2}`
`<=>y={2m-4m^2+4+4m^2}/{1+m^2}`
`<=>y={2m+4}/{1+m^2}`
Ta có: `S=x+y`
`<=>S={1-2m}/{1+m^2}+{2m+4}/{1+m^2}`
`<=>S={1-2m+2m+4}/{1+m^2}=5/{1+m^2}`
Ta có: `m^2\ge 0\ \forall m`
`<=>1+m^2\ge 1`
`<=>5/{1+m^2}\le 5`
`<=>S\le 5 \ \forall m`
Dấu "=" xảy ra khi $m=0$
Vậy $S=x+y$ có $GTLN$ bằng $5$ khi $m=0$.
