Thank you for your answer. I study maths in Lithuanian, so I don't see English exercises often. It was written in Lithuanian. I can translate the task from LT to EN. It sound like that "Which what x and y values this is a correct equality?"
My point was not that the question was written badly, but that it is surprising! Normally one would not solve one equation for both variables, but in this case it can be done, as you discover when you do the work of trying to graph the equation (as Khan implied and I stated explicitly).
So what you're saying is that I should make a "function"
(That is how we call it in LT, I don't know how it is called in English).I find what y is equal to, then I choose 2 or more y's and calculate the x's. Then I make a graph and see where it intersects. I seem to get the concept, but I get stuck at trying to solve for y. I can't part x from y. Even if I get that "y=..." in that "..." there is y too, and it also
looks really wrong.
Here is where I go out of ideas.
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The word "function" is correct in English; "solving for y" is the same as "expressing y as a function of x".
But I don't understand what you are saying you describe here, "choose 2 or more y's and calculate the x's".
The work in the first link is correct until you started trying to solve for y. You have, in effect,
2x^2 + 2y^2 - 2xy - 2y + 2/3 = 0
But you did not write it as a quadratic
in y, which is why you still had y in the result. I would rearrange the equation as
2y^2 - (2x + 2)y + (2x^2 + 2/3) = 0
so that a = 2, b = 2x + 2, and c = 2x^2 + 2/3. Then you will be using the quadratic formula and looking at the discriminant, in particular.