Restoring and balancing an equation

This lesson asks you to solve equations that involve multiple terms, integers and decimal numbers in various contexts, and to verify your solutions. The museum offers two cards, on the science of algebra and on the types of equation in al-Khwarizmi's Book of al-Jabr and al-Muqabala; solving the equations of this grade is for your class. ¹

Write an equation with terms on both sides, then add the same quantity to both sides or remove the matching term from both. At each step check that the equation is still true for the solution. Then put the solution back into the original equation.

Al-Khwarizmi shows the two operations of algebra in the phrases of his problems. In his problems al-jabr is completing a quantity that has a deficiency or deduction by adding its like to both sides, as in “restore the thirty by the thing and add it to the two things” ². Al-muqabala is casting away like terms from both sides, as in “you cast away a square by a square” ³. In his worked problems, as the record of his algebra describes them, the restoring step appears again and again: a quantity “less ten things” is restored and the ten things are added to the other side ⁴.

His preface, as the record of his algebra reports it, praises God and asks blessings on the Prophet ﷺ Muhammad, then says what moved him to write ⁵. The caliph al-Ma'mun loved learning, and this encouraged him ⁵. He limited the book to what people need constantly: inheritance, bequests, the division of estates, judgements, trade, land measurement, digging canals, engineering, and other matters ⁶. A long Book of Bequests in the same work applies its rules to shares of estates and bequests ⁷.

Where does the word jabr come from? The Encyclopaedia of Islam, in its article on al-Khwarizmi, reports Gandz's proposal that it comes from Assyrian gabru, and then notes a difficulty: the word is not attested in any intervening language such as Greek ⁸. The article judges it more likely that the word comes from medical vocabulary, where jabr means setting a dislocated limb ⁸. The record reports both and decides nothing.

Al-Khwarizmi says what his book is for, and the encyclopaedia proposes where its key word came from. Which of the two would you trust to tell you why the book was written, and why?

• Algebra (al-jabr): the two operations in his own words.
• The six types of equation: a worked example for each type.
• The Book of al-Jabr and al-Muqabala: the preface, the chapters and the Book of Bequests.
• al-Khwarizmi: what is known of his life, and the question of the word jabr.

Read about algebra and the six types. Choose one example of the six types, solve it your own way, and verify the answer by substitution.