Verified answer

The two lines are not parallel, so they intersect.

The [minimum] distance is zero.

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Distance as a function of x at fixed y:

d(x) = abs(y1 - y2)

= abs( 0.5 (-1-3x) - 0.25 (-12 - 3x) )

= 0.25 abs( -2 (1+3x) + (12 + 3x) )

= 0.25 abs( -2 - 6x + 12 + 3x )

= 0.25 abs( 10 - 3x )

Distance as a function of y at fixed x:

d(y) = abs(x1 - x2)

= abs( (1/3)(-1-2y) - (1/3)(-12-4y) )

= (1/3) abs( -1 - 2y + 12 + 4y )

= (1/3) abs( 11 + 2y )