How did the Egyptian Rhind papyrus use an early root-finding method?

The Rhind papyrus did not use root-finding methods in the modern sense of calculating square or cube roots; rather, it employed a technique known as the method of false position (or false assumption) to solve linear equations in one unknown.12

In this method, the scribe would replace the unknown quantity with a convenient "false" trial value to see how it affected the equation. By comparing the result of this trial calculation to the actual target value, the scribe could determine the correct answer through a simple proportional adjustment.1

For example, in a problem asking for a quantity such that the value plus its seventh part equals 19, a scribe might first assume the quantity is 7. Since 7 plus its seventh (1) equals 8—not 19—the scribe would then multiply the trial value (7) by the ratio of the target (19) to the result (8) to find the correct solution. This approach is a foundational arithmetic procedure for solving equations where the unknown is only to the first power.123