Here is my workflow for your last example. It is not the best workflow, but how I approached it.
Visualize the first two equations:
planes =
ContourPlot3D[{x + y + z == 1, x - 2 y - z == 3}, {x, -2, 2}, {y, -2,
2}, {z, -2, 2}]
Look for a location where I might visualize the third equation:
FindInstance[(y - x^2)^2 + (z - x^3)^2 == 0, {x, y, z}, 6] // N
See if I can find conditions that there are real solutions to the last equation:
Reduce[(y - x^2)^2 + (z - x^3)^2 == 0, {x, y, z}, Reals]
Continue to see if there might be any real solutions other than 0:
Reduce[-x^4 + 2 x^2 y - y^2 > 0, {x, y}, Reals]
I conclude there is no intersection between the three surfaces.
For your second example:
eq1 = x y - z^2 - x;
eq2 = x y z - 2;
ineq3 = x^2 + y^2 + z^2 <= 5;
Sanity check to see if there are any intersections:
FindInstance[{eq1 == 0, eq2 == 0, ineq3}, {x, y, z}, Reals, 6] // N
Reduce[{eq1 == 0, eq2 == 0, ineq3}, {x, y, z}, Reals] // N
It looks like there is a narrow band around x=-1 that may have solutions in it.
Plot the first two surfaces in the region of interest:
surfaces =
ContourPlot3D[{eq1 == 0, eq2 == 0}, {x, -1.5, -1}, {y, -10,
10}, {z, -10, 10}]
plot the inequality region:
region =
RegionPlot3D[ineq3, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
PlotStyle -> {Opacity[0.5], Orange}] (*I added Opacity on a second pass, not on the first work flow*)
VIsualize the intersection:
Show[surfaces, region]
Look for the curve. Copying and pasting as this is a one-time-only:
pp = ParametricPlot3D[{x, Root[-4 - x^3 #1^2 + x^3 #1^3 &, 1],
2/(x Root[-4 - x^3 #1^2 + x^3 #1^3 &, 1])}, {x, -1.15, -1.06},
PlotStyle -> {Red, Thickness[0.02]}]
Visualize:
Show[surfaces, region, pp]