How can one fourthx − 3 = one halfx + 8 be set up as a system of equations?

First, set each half of the equation equal to zero:

1/4x - 3 = 0
1/2x +8 = 0

Notice how I stacked the equations, lining up "like terms."

Multiply both equations by "4" to eliminate the fractions:

(4) 1/4x -3 = 0 -->    x -  12 = 0
(4) 1/2x+8 = 0 -->  2x + 32 = 0

Add the equations equal:                3x + 20 = 0
Subtract 20 from both side                   - 20  -20 
Simplify                                           3x         = -20
Divide both sides by 3                    3                 3
Final x = -20/3