$\left \{ {{2x+5y=7m+2} \atop {2x+3y=m+2}} \right.$
`<=>`$\left \{ {{2x+5y-2x-3y=7m+2-m-2} \atop {2x+3y=m+2}} \right.$
`<=>`$\left \{ {{2y=6m} \atop {2x+3y=m+2}} \right.$
`<=>`$\left \{ {{y=3m} \atop {2x+3.3m=m+2}} \right.$
`<=>`$\left \{ {{x=-4m+1} \atop {y=3m}} \right.$
Ta có:
$2x+y=5$
`<=>`$2(-4m+1)+3m=5$
`<=>`$-8m+2+3m=5$
`<=>`$-5m=3$
`<=>`$m=-\frac{3}{5}$
Vậy $m=-\frac{3}{5}$ thỏa mãn yêu cầu bài toán
