How did the authors of the constitution hope the House of Representatives and senate would balance each other

By the system of self-checking.

Explanation:

Balance in U. S. constitution is all about the Separation of Powers, how the power is shared between the branches and how it can be challenged by one branch on one another.

The Legislature branch is a bicameral system, that combined the ideas that each state should have equal representation in the federal government (Senate) and representatives that were based on population number (House of representatives).

To guarantee the balance of House of Representatives and Senate a system of self-checking was created that includes:

* Bills must be passed by both houses of Congress;

* House must originate revenue bills;

* Neither house may adjourn for more than three days without the consent of the other house;

* All journals are to be published.