Recently, a novel flexible-antenna technology, called pinching antennas, has attracted growing academic interest. By inserting discrete dielectric materials, pinching antennas can be activated at arbitrary points along waveguides, allowing for flexible customization of large-scale path loss. This paper investigates a multi-waveguide pinching-antenna integrated sensing and communications (ISAC) system, where transmit pinching antennas (TPAs) and receive pinching antennas (RPAs) coordinate to simultaneously detect one potential target and serve one downlink user. We formulate a communication rate maximization problem subject to radar signal-to-noise ratio (SNR) requirement, transmit power budget, and the allowable movement region of the TPAs, by jointly optimizing TPA locations and transmit beamforming design. To address the non-convexity of the problem, we propose a novel fine-tuning approximation method to reformulate it into a tractable form, followed by a successive convex approximation (SCA)-based algorithm to obtain the solution efficiently. Extensive simulations validate both the system design and the proposed algorithm. Results show that the proposed method achieves near-optimal performance compared with the computational-intensive exhaustive search-based benchmark, and pinching-antenna ISAC systems exhibit a distinct communication-sensing trade-off compared with conventional systems.