First, we should find the coordinates for the x and y intercepts for the equation 3x - 6y = 12 + 8x - 3y.

x-intercept:

The x-intercept is the point at which the line passes through the x-axis. This means that y is always equal to zero. Thus, we can set the y values in the equation to zero. 3x - 6(0)= 12 + 8x - 3(0). Solving for x yields -12/5, so our x-intercept is at (-12/5, 0)

y-intercept:

The process is the same as for the x-intercept, except it is the x-value this time that is 0. 3(0) - 6y = 12 + 8(0) -3y. Solving for y yields a value of -4. Thus, our y-intercept is at (0, -4)

To find the distance between our ordered pairs of (-12/5, 0) and (0, -4), we use the distance formula: \(d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }\) where our ordered pairs represent (x1, y1) and (x2, y2)

Plugging our variables into the formula gets our final answer of d = (4 sqrt(34))/5 ≈ 4.66476