I don't get the same graph, before and after solving for Y?

Now, if I draw the following:
$3x+y=3$
$2x^2-y^2=-1$
With Wolfram I get the following graph.

But if I draw the following functions that I have solved for Y
$y=\sqrt{1+2x^2}$
$y=3-3x$
I seem to be loosing some information

Is all this simply a consequence of the following math:
$y^2=1+2x^2$
$y=\sqrt{1+2x^2}$

Squaring could cause this, but is that what I have done here? The reason for my post is that when doing the calculations by hand I still find the two intersections. But it is not visible in the graphs of the very same functions that on paper gives me what I want.

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You're right. To complete things, plot the negative square root as well. –  Ｊ. Ｍ. Oct 4 '11 at 22:43
Note, for example, that $(0,-1)$ is on the graph of $y^2=1+2x^2$ but not on the graph of $y=\sqrt{1+2x^2}$. When you say you get both intersections by hand, surely that's because somewhere along the way you "square both sides" thereby re-introducing the other branch of the hyperbola. –  Gerry Myerson Oct 4 '11 at 22:56
Remember that $\sqrt{y^2} = |y|$, not $y$. So your first function should not be $y=\sqrt{1+2x^2}$, but rather $|y|=\sqrt{1+2x^2}$. –  Arturo Magidin Nov 28 '11 at 4:32

The "missing information" comes from improperly taking the square root. The actual equation B, solved for y, is: $$y \ =\ \pm\ \sqrt{1 \ +\ 2x \ ^2\ }$$ This will give you the second part of the graph that you were previously missing.