For the second equation you gave, there's a very odd break in the line. If you zoom towards the origin of the graph (try Xmin=-0.5, Xmax=0.8, Ymin=-1.1, Ymax=0.4), you can see it more clearly. You'll see that right before the break, the second equation curves up like a parabola (then disappears) before curving down after the break.
The first equation you gave is just a horizontal line, but since it's so small, you can treat it as if it were the x-axis.
My guess is that when you went to [2nd][Trace](Calc) > 5:intersect, you didn't specify the appropriate bounds.
The left-bound should be on the horizontal line, somewhere to the left of the y-axis. The right-bound should be on the second line, somewhere to the right of the y-axis, but before the gap in between. The guess is somewhere in between those two bounds.
If you do this, then the calculator returns an "accurate" result. If you place the right bound on the second line AFTER the gap, then it will give a Err:No Sign Chng because they never truly intersect. Nothing's wrong with your TI-83+, it's just that you specified different bounds for it.