Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In elementary algebra, those symbols (today written as Latin and Greek letters) represent quantities without fixed values, known as variables. Just as sentences describe relationships between specific words, in algebra, equations describe relationships between variables.

I have two fields that total 1,800 square yards. Yields for each field are ⅔ gallon of grain per square yard and ½ gallon per square yard. The first field gave 500 more gallons than the second. What are the areas of each field?

It's a popular notion that such problems were invented to torment students, and this might not be far from the truth. This problem was almost certainly written to help students understand mathematics — but what's special about it is it's nearly 4,000 years old! According to Jacques Sesiano in "An Introduction to the History of Algebra" (AMS, 2009), this problem is based on a Babylonian clay tablet circa 1800 B.C. (VAT 8389, Museum of the Ancient Near East). Since these roots in ancient Mesopotamia, algebra has been central to many advances in science, technology, and civilization as a whole. The language of algebra has varied significantly across the history of all civilizations to inherit it (including our own). Today we write the problem like this:

x + y = 1,800

⅔∙x – ½∙y = 500