If 4x +2y=200 and 2x+7y=400 how do you solve for x and y

This is what you call systems of equations. Basically were the two equations cross on a graph is the solution
 
You can solve this multiply ways... Graph, table, Substitution, elimination...

I will choose elimination because it is the easiest for how the equations are set up.

So

4x+2y = 200     2x+7y=400

First I take the second equation and multiply it by 2..
2(2x) + 2(7y)= 2(400)

4x+14y=800 

Now I can just subtract the two equations

  4x+2y = 200
-(4x+14y=800)

-12y=-600

Solve for y

-12y/-12=-600/-12

y=50

Plug back in

4x+2y = 200

4x+2(50)=200
4x+100=200
     -100  - 100
4x=100

4x/4=100/4

x=25

(25,50)

To check if you are correct you can use something like desmos (online graphing tool) and see if (25,50) is were the two lines cross.

You can also use substitution if you prefer.