Ancient Egypt
Rhind Papyrus
Perhaps the oldest known record of linear equations, the Rhind Papyrus from 1650 BC documented the processes Egyptians used to solve linear equations. Just like the Babylonians, Egyptians communicated problems verbally.
(Unit) Fractions
When making calculations involving non-integer variables and solutions, Egyptians would express the answer as a fraction. However, they only used unit fractions – all of the fractions they used always had one as the numerator. For example, they would express ⅚ as ½ + ⅓.
False position (Regula Falsi)
To solve linear equations, Egyptians used a process now known as False position. Essentially, they would note down an equation for an unknown quantity, where the left included all terms with unknowns, and the right would be a number. They would then plug in a number for all the terms with the unknown quantity that would make the output easy to compute. If the output was less than the right number by a factor of x, they would increase the "guess" by x. (e.g. x + ⅕x = 18 => plugin in 5, you get 6 which is 3x less than 18, so the answer is x=5*3=15). The images on the left graphically and algebraically show how the algorithm is applied today.