How does an equation that's not fully factored, end up outputting solutions?

Question

In a textbook I stumbled across this:

$(32-8a+2b)x+(32-6a+b)=0$
Thus, $32-8a+2b=0$ (1) , $32-6a+b=0$ (2)

Now how exactly does that first equation, imply the 2 solutions? In this case the LHS doesn't consist of factors only. So how are these the solutions?

Answer 1

What you write doesn’t allow you to write that both coefficients are equal to zero. However, had you used correct mathematical symbols such as $\forall{x}, ax+b =0 $, you could conclude that $a=0$ and $b=0$

Answer 2

Since, the equation $mx + c = 0 $ should hold for every $x$ then it makes coefficients $m, c$ equal to zero due to the fundamental theorem of algebra.