How to solve a system of two equations with two unknowns?

Question

So i'm working through a Cournot duopoly problem. Here is the answer sheet with the question and the explanation of the various steps.

Im perfectly fine finding q1 and q2 but im having problems getting to the answer 8.11)

How do i find that solution? What method do i need to use ? Could somebody please explain? How is this method called?

I tried subsituting q2 into q1 but it gets really messy.

Thanks!

EDIT:

Answer 1

You could rewrite (8.9) and (8.10) as

$$2bq_1= A -bq_2-c$$ $$2bq_2= A -bq_1-c$$

Subtract the two equations from each other,

$$ 2b(q_1-q_2)= b(q_1-q_2)$$

or $b(q_1-q_2)=0$, which yields $q_1=q_2$. Plugging this back to the two equations above, you get $3q_1=3q_2=A-c$, or

$$q_1=q_2=\frac{A-c}{3b}$$

Edit: Plug (8.9) into (8.10)

$$q_2=\frac{ A -b\frac{ A -bq_2-c}{2b}-c}{2b}$$ $$=\frac{ A -\frac12( A -bq_2-c)-c}{2b}$$ $$=\frac{ \frac12 A +\frac12 bq_2-\frac12c}{2b}=\frac{ A + bq_2-c}{4b}=\frac14q_2+\frac{ A-c}{4b}$$

Then, move $\frac14q_2$ to the LHS to get $$q_2-\frac14q_2=\frac{ A-c}{4b}\implies \frac34q_2=\frac{ A-c}{4b}\implies q_2 =\frac{ A-c}{3b} $$