If |x| + |y| = 10 and x = 6, what are the possible values of y?

Substitute $x=6$: $|6|+|y|=10$, so $6+|y|=10$ and $|y|=4$. Both $4$ and $-4$ have an absolute value of $4$, so $y=4$ or $y=-4$, choice D. Choices A and C each give only one of the two values. Choice B ($y=\pm16$) comes from using $|y|=10+6$ instead of $10-6$. Check: $|6|+|4|=10$ and $|6|+|-4|=10$. Whenever an absolute value is set equal to a positive number, expect both a positive and a negative solution. Substituting both values back confirms they work.