The general curve of a system which is linearly dependent on a certain number of given irreducible curves will not have a singular point which is not fixed for all the curves of the system.
Bertini's Theorem
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References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 115, 1959.Referenced on Wolfram|Alpha
Bertini's TheoremCite this as:
Weisstein, Eric W. "Bertini's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BertinisTheorem.html