Question

Item 6 for what value of a is 8x−8+3ax=5ax−2a an identity?

Answer 1

Let's rewrite the expression as

$8x+3ax-5ax = 8-2a \iff 8x - 2ax = 8-2a \iff 2x(4-a) = 2(4-a)$

So, if $a \neq 4$, you may divide both sides by $4-a$, the equation becomes $2x=2$, and the only solution is $x=1$, which means that this is not an identity (an indentity is tautologically true, no matter the value of x).

If instead $a = 4$, you are multiplying both sides by zero, so the equation becomes

$2x \cdot 0 = 2 \cdot 0 \iff 0=0$

So, it doesn't matter which value for x you choose, because you will always end up with $0=0$, which is obviously always true.