Use elimination
x+5y=-2
2x + 5y = 6

[multiply the first equation by -2]
[-2x - 10y = 4]
[add like terms between the equations]
[-5y = 10]
[y = -2]
[x = 8]
(8, -2)

You can use the subtraction method to eliminate the 5y. Then you would subtract x and 2x to get -x. Next, subtract the 6 from the -2 to get -8. Now we know that -x=-8 so then x=8. Finally substitute x for 8 and solve for y