Deployment of machine-type communications (MTCs) over the current cellular network could lead to severe overloading of the radio access network of Long Term Evolution (LTE)-based systems. This paper proposes a slotted access-based solution, called the Slotted Access For Group Paging (SAFGP), to cope with the paging-induced MTC traffic. The proposed SAFGP splits paged devices into multiple access groups, and each group is then allocated separate radio resources on the LTE's Physical Random Access Channel (PRACH) in a periodic manner during the paging interval. To support the proposed scheme, a new adaptive barring algorithm is proposed to stabilize the number of successful devices in each dedicated access slot. The objective is to let as few devices transmitting preambles in an access slot as possible while ensuring that the number of preambles selected by exactly one device approximates the maximum number of uplink grants that can be allocated by the eNB for an access slot. Analysis and simulation results demonstrate that, given the same amount of time-frequency resources, the proposed method significantly improves the access success and resource utilization rates at the cost of slightly increasing the access delay compared to state-of-the-art methods.