Summary: We consider linear iterated function systems with a random multiplicative error on the real line. Our system is \(\{x\mapsto d_i+\lambda_iYx\}_{i=1}^m\), where \(d_i\in\mathbb R\) and \(\lambda_i>0\) are fixed and \(Y>0\) is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors \(y_1,y_2,\dots\), distributed as \(Y\), independent of everything else. Let \(h\) be the entropy of the process, and let \(\chi=\mathbb E[\log(\lambda Y)]\) be the Lyapunov exponent. Assuming that \(\chi<0\), we obtain a family of conditional measures \(\nu_y\) on the line, parametrized by \(y=(y_1,y_2,\dots)\), the sequence of errors. Our main result is that if \(h>|\chi|\), then \(\nu_y\) is absolutely continuous with respect to the Lebesgue measure for almost every \(y\). We also prove that if \(h<|\chi|\), then the measure \(\nu_y\) is singular and has dimension \(h/|\chi|\) for almost every \(y\). These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.