Solve for real numbers x, y, and z
Graph the 3 equations with Graphing Calculator as 3D. The image shows three intersecting tubes with elliptical cross-sections. Click here to open a GCF file. (You must have Graphing Calculator 3.5 to open this file but the image could be graphed with other software.) Opening in GC 3.5 allows the image to rotate ans show different perspectives.
Examining the 3D image will not provide a solution but it may suggest where to look. For example it appear the three tubes may have a common point in the region where x, y, and z are positive.
Try fixing one of the variables -- for example replace z with k -- and examine some 2D graphs for different k. These are contour plots for each z = k plane. One solution is very close to (x, y, x) = (1.71674, 3.05009, 5.679).