$2x^2 + 3y^2=0$. This is possible only when both the value of $x$ and $y$ are zero. But the thing is I fail to understand the significance of this equation. Why would such an equation exist or why we create them?
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3I assume that you intend $2x^2+3y^2=0$. The only pair $(x,y)$ of reals at which this holds is indeed $(0,0)$. One usually doesn't start from such an uninteresting equation. But sometimes we bump into it during a computation. – André Nicolas Jun 27 '13 at 02:48
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As you note, the only solution to $\;2x^2 + 3y^2 = 0$, is indeed $(x, y) = (0, 0)$. It is not a very interesting question, in itself. No one with the least bit of creativity would simply create the equation to pose as an exercise: few teachers or texts would do so, I imagine.
But it may happen that something of the sort "crops up" in the process of solving a more complex equation, after simplifying/canceling, or when searching for "zeros."
If you did mean to write $2x^2 + 3x^2 = 5x^2 = 0 \implies x = 0$, that might be even less interesting of an equation, but may crop up when finding the zero's of, say, $f(x) = 5(x^4 - x^2) = 5x^2(x^2 - 1) = 5x^2(x-1)(x+1),\;$ which is not too terribly interesting, but the point is, many exercises are created simply to test one particular concept or technique, as this slightly more complicated function might be used.
amWhy
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Interpreting this equation using analytic geometry tells us something. The equation $$2x^2+3y^2=r^2$$ with $r>0$ is an ellipses with center at $(0,0)$. As $r$ shrinks to 0, the ellipse gets smaller and smaller, until finally when $r=0$ it shrinks to a point. People sometimes say that $2x^2+3y^2=0$ gives a degenerate ellipse.
We can also graph the equation $z=2x^2+3y^2$ in three dimensions. The graph is a kind of bowl, wider along the x-axis than along the y-axis. If we slice it with a horizontal plane $z=k$, we get the ellipse $2x^2+3y^2=k$, provided $k>0$. When $k=0$, the horizontal plane is just the xy-plane, which is tangent to the bowl at the single point (0,0,0). And when $k<0$, the plane and bowl don't intersect at all.
In general, when people study equations (or systems of equations) in several variables, they often look at whole families of equations: that is, they include one or more parameters in the equation (like $k$ or $r$ above). At certain specific values of the parameters, special things can happen. The study of those special values is important in several branches of mathematics and physics.
Another example you may find interesting: $x^2-y^2=k$. When $k>0$, we have a hyperbola that opens out to the right and the left. When $k<0$, we have a hyperbola opening out above and below. Can you tell what we get when $k=0$? (Hint: factor $x^2-y^2$.)
Michael Weiss
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