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Identify the solution(s) of the system of equations, if any

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By the basic principles of math, we know that we are allowed to add equal amounts to both sides of an equation without changing it. 
i.e. if we have x = 4, we also know that x + 2 = 6 is true because we added the same amount to both sides.
In this case, we know that 6x + 2y = 6. That is, the amount (6x + 2y) is the same amount as (6). Conveniently, we see that in the first equation, there are also two y's. Since there are an equal number of y's in the top and bottom equations, we know that if we subtract the two the y's will cancel and we will be left with only one variable, x. This is a valid operation because we are subtracting the same amount from each side of the equation (as described above). Thus we get:
 (7x + 2y) = 2
-(6x + 2y) = 6
    x + 0    = -4
in other words: x = -4. Since we know x's value, we can plug it back into the equation to get a value for y. Once you have that y value, don't forget to check your work by plugging the two numbers back into the second equation just to make sure that you have the right answer. 
A second way to do this (probably the more straight forward method), is to solve the first equation for one of the two variables. Let's choose x! After subtracting over the y and dividing through by 7 we have:
x = (2 - 2y)/7
now lets plug this value into the second equation:
6*(2 - 2y)/7 + 2y = 6
Now you have an equation with only one variable, so you can solve for the value of y by doing a few multiplications and additions. Once you find y, plug it back into the equation to get the value of x, again checking your work by making sure the two numbers you find work for both of the equations.