You are right Tim, it's easy to get configuration that show 4 solutions, so equations may have no analytical solutions. Fiddling with Geogebra, I came to the kind of configuration below, where the 5th bar tip circle is more or less tangent to the locus of 4 bars linkage. Physically, with the play in joints, you get something no longer static, that can move in some range...
Static 5 bars linkage angles calculation
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