7x + 2y =10
-7x+y=-16
using elimination method

First, I would write the problem vertically because it just helps me better.

7x+2y=10
-7x+1y=-16  

Then, I would add downwards.

    7x+2y=10
+ -7x+1y=-16

    0x+3y=-6

Now, divide 3 from both sides.

3y=-6
---  ---
3     3

y=-2 

Now, plug in -2 for y in any equation to get x.

7x+2(-2)=10

Combine like terms

7x-4=10

Add 4 to both sides

7x-4=10
   +4  +4

7x=14

Divide each side by 7.

7x=14
 ---
7    7

x=2 

So your solution is
(2,-2)

To check the solution, plug in the x and y values in both equations.

7(2)+2(-2)=10
14-4=10
10=10 

-7(2)-2=(-16)
-14-2=-16
-16=-16 

Therefore, the solution works.
You must always plug in the x and y values into both equations to check the solution because sometimes the solution will work for one and not the other.
The solution must work to both equations in other for it to be right.