How to predict one equilibrium constant given two others? $$\begin{alignat}{2}
\ce {CO2 (g) + 3H2 (g) \;&amp; &lt;=&gt; CH3OH (g) + H2O (g)}\qquad&amp;&amp;{k_1=\;?} \\
\ce {CO (g) + H2O (g) \;&amp; &lt;=&gt; CO2(g) + H2(g)}\qquad&amp;&amp;{k_2= 1.0\times10^5} \\
\ce {CO (g) + 2H2 (g) \;&amp; &lt;=&gt; CH3OH(g)}\qquad&amp;&amp;{k_3= 1.4\times10^7}
\end{alignat}$$

What is the value of $k_1$? I have tried reversing equation two and then multiplying $k_2$ and $k_3$ together, but I am not sure how to work this problem out.
Remember that when you reverse the reaction the equilibrium constant changes. For the general gas-phase reaction $\ce{A(g) + B(g) &lt;=&gt; C(g) + D(g)}$ the equilibrium constant expression is $$K_\text{f} = \frac{p_\ce{C} p_\ce{D} }{p_\ce{A} p_\ce{B}}$$ (Strictly, it's activities, not partial pressures, but the principle is the same.)
The reversed reaction $\ce{C(g) + D(g) &lt;=&gt; A(g) + B(g)}$ has the equilibrium constant expression $$K_\text{r} = \frac{p_\ce{A} p_\ce{B} }{p_\ce{C} p_\ce{D}} = \frac{1}{K_\text{f}}$$

You are correct in multiplying the equilibrium constants to get the equilibrium constant for the combined reaction.