The absolute value of any real number is non-negative. Therefore, for the sum of two absolute values to be equal to zero, both individual absolute values must be equal to zero.

This means that:

x + 12y + 19 = 0
x + 2y - 1 = 0

Solving these two equations simultaneously, we get:

x = -12y - 19
x = -2y + 1

Setting these two expressions equal to each other, we get:

-12y - 19 = -2y + 1
-10y = 20
y = -2

Substitute y = -2 back into one of the equations:

x = -12(-2) - 19
x = 24 - 19
x = 5

Therefore, the solution to the equation |x+12y+19| + |x+2y-1| = 0 is x = 5, y = -2.