Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements. 
Let  $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. 
I am working on a research project, where I bounded a certain expectation $E$ by
$E \ll S + S'$, where
$$
S = \sum_{ \substack{   x_i \in \mathbb{F}_q[t] \backslash \{ 0 \} \ (1 \leq i \leq 6)   \\ x_1 + x_2 + x_3 =n \\ x_1 + x_4 = x_5 + x_6 } }
q^{ -\gamma (\deg x_1 + \deg  x_2 + \deg x_3 + \deg x_4 + \deg  x_5 + \deg x_6 )  },
$$
$$
S' = \sum_{ \substack{  x_i \in \mathbb{F}_q[t] \backslash \{ 0 \} \ (i = 1,2,3,5,6 )    \\ x_1 + x_2 + x_3 =n \\ x_1 + x_2 = x_5 + x_6  } }
q^{ -\gamma (\deg x_1 + \deg  x_2 + \deg x_3 + \deg  x_5 + \deg x_6 )  }.
$$
I managed to find a bound for $S$ and $S'$ by "brute force" and found a bound for $E$.
What I Was wondering was is there possibly an easy or obvious way to see that $S' \ll S$?
The reason why I ask is I had many cases to consider and I am looking for a way to cut down on the amount of computation I wrote up (and many of the cases look like what I have here). I would greatly appreciate any advice. Thanks! 

ps Let $\gamma$ be a positive real number sufficiently large so that both sums converge.