This paper deals with some theoretical and numerical results for Volterra Integral Algebraic Equations (IAEs) of index-1 with weakly singular kernels. This type of equations typically has solutions whose derivatives are unbounded at the left endpoint of the interval of integration. For overcoming this non-smooth behavior of solutions, using the appropriate coordinate transformation the primary system is changed into a new IAEs which its solutions have better regularity. An effective numerical method based on the Chebyshev collocation scheme is designed and its convergence analysis is provided. Our numerical experiments show that the theoretical results are in good accordance with actual convergence rates obtained by the given algorithm.