Green’s Dictionary of Slang

slate n.1

[SE slate]

1. [mid-16C–mid-19C] a bedsheet.

2. [mid-19C+] one’s bill, the credit one has run up; usu. in a public house or bar.

In phrases

clean (off) the slate (v.) (also wipe off the slate)

[mid-19C+] lit. or fig., to pay off one’s outstanding debts.

on one’s/the slate [the practice of writing public house debts on a slate]

[late 19C+] on credit.