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So I have several descriptions of reactions and I have to give the net ionic equation. Are these just the equations or are they the proper net ionic ones?

Solid calcium is added to warm water. $$\ce{Ca_{(s)} + 2 H_{2}O_{(l)} -> Ca^{2+}_{(aq)} + 2 OH_{(aq)}^- + H2_{(g)}}$$

Powdered magnesium oxide is added to a container of carbon dioxide gas. $$\ce{MgO_{(s)} + CO2_{(g)} -> MgCO3_{(s)}}$$

Did I do those right, or am I on the wrong track?

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Purpose of ionic equation is to describe ions in (aqueous) solution, without the need to bother about details.

The first equation looks reasonable, you could possibly end up with $\ce{Ca(OH)_2}$, which is not very water soluble, but it's IMHO the type of detail you should not bother with.

The second one is more difficult, though. At first, there is no solution to speak about, therefore the request for ionic equation itself is invalid. The reaction written is correct, as long as it describes possibility. If it would really happen is question of reaction conditions, therefore totally outside the scope of homework. As such, I'd say, correct.

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