Given that the line $y = ax + b$ minimizing the least squared error (LSE) is found by solving the system of equations
$$ \big(\sum x_i^2 \big{)}a + \big(\sum x_i \big{)}b = \sum x_iy_i $$ $$ \big(\sum x_i \big{)}a +nb = \sum y_i $$
I've been trying to show whether or not the value of $a$ stays the same upon adding a constant $c$ to all points $y_i$. That is, if $(a_c, b_c)$ satisfy
$$ \big(\sum x_i^2 \big{)}a_c + \big(\sum x_i \big{)}b_c = \sum x_i(y_i + c) $$ $$ \big(\sum x_i \big{)}a_c +nb_c = \sum (y_i + c) $$
then $a_c = a$.
I've solved for $a$ to find $$ a = \frac{(\sum y_i)(\sum x_i y_i) + n\sum x_i y_i}{(\sum x_i)(n \sum x_i + \sum x_i y_i)} $$
But am having a hard time showing whether $a = a_c$ from here.
Your formula for $a$ is not correct. It should be $$a=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2}.$$
Now if you add $c$ to all the $y_i$, you get $$a_c=\frac{n\sum x_i(y_i+c)-\sum x_i\sum (y_i+c)}{n\sum x_i^2-(\sum x_i)^2}.$$
See if you can expand and cancel to get the same result from here.