Introduction:
The accounting equation, also known as the balance sheet equation, is a fundamental principle in accounting that states that assets must equal liabilities plus equity. It can be expressed as follows: Assets = Liabilities + Equity. This equation holds true under all circumstances and is the foundation for double-entry bookkeeping.

Explanation:

1. Illustration 1:
Let's consider a scenario where a business owner invests $10,000 in cash to start a new business. The accounting equation can be applied to this situation as follows:

- Assets: The cash received from the owner is considered an asset. So, the asset side of the equation increases by $10,000.
- Liabilities: Since no loans or debts are involved, the liabilities side of the equation remains unchanged at $0.
- Equity: The owner's investment is considered equity. Therefore, the equity side of the equation increases by $10,000.

After this transaction, the accounting equation still holds true:
Assets ($10,000) = Liabilities ($0) + Equity ($10,000)

2. Illustration 2:
Now, let's consider a different scenario where a business purchases equipment for $5,000 by taking a loan of $3,000 and paying $2,000 in cash. The accounting equation can be applied as follows:

- Assets: The equipment purchased is considered an asset. So, the asset side of the equation increases by $5,000.
- Liabilities: The loan taken for the equipment is considered a liability. Therefore, the liabilities side of the equation increases by $3,000.
- Equity: The cash paid by the business owner is considered equity. So, the equity side of the equation decreases by $2,000.

After this transaction, the accounting equation still holds true:
Assets ($5,000) = Liabilities ($3,000) + Equity ($3,000)

Conclusion:
The accounting equation holds good under all circumstances as it represents the fundamental principles of accounting. It ensures that the balance sheet remains in balance, providing a clear and accurate snapshot of a company's financial position. These illustrations demonstrate how the equation is applicable in different scenarios, emphasizing its consistency and accuracy in reflecting the financial state of a business.