Identifying a system's breakpoint in performance testing is significant because it helps in understanding the limitations of the system. It represents the point at which the system fails or becomes unstable under a specific load. By identifying the breakpoint, testers can determine the system's capacity, scalability, and potential bottlenecks. This information is crucial for optimizing system performance and ensuring it meets the desired requirements under varying workloads.

Apache JMeter is a performance testing tool that is effective for load balancing testing by simulating concurrent user load on a web application. It allows testers to analyze the application's performance under different levels of load and helps identify potential bottlenecks in the system. By generating realistic user scenarios, JMeter can provide valuable insights into how well the application handles simultaneous requests and whether the load is distributed evenly across servers.

In Agile methodology, automation tests are conducted before each sprint to ensure that the developed features meet the acceptance criteria and are free from defects. Automated tests help in achieving continuous integration and delivery by providing rapid feedback on the application's functionality. This frequent execution of automated tests aligns with Agile principles, allowing teams to catch issues early in the development process and deliver high-quality software iteratively.

Postman's use of Environment Variables as an advanced feature allows for dynamic data use in API testing. Environment Variables in Postman enable testers to manage and reuse data across multiple requests, enhancing flexibility and efficiency in API testing scenarios where dynamic data is crucial for thorough testing.