What could go in the blank for - (x-8) + 4x=2 (__) + x with infinitely many solutions?

To make the equation into an infinitely many solution, the coefficient and variable on each side must be the same. Let's see how it goes.

- (x - 8) + 4x = 2 (__) + x

-x + 8 + 4x = 2 (__) + x

3x + 8 = 2 (__) + x

3x - x + 8 = 2 (__)

2x + 8 = 2 (__)

(2x + 8) / 2 = (__)

x + 4 = (__)

Therefore, the answer of the blank will be x + 4.

Let's verify.

- (x - 8) + 4x = 2 (x + 4) + x

-x - 8 + 4x = 2x + 8 + x

3x + 8 = 3x + 8

As you can see each side has the same coefficient and variable which is 3x. Once you subtract them, you will get zero. You can't solve for x as well.