As knowledge about electric circuits was first developing, in 1848 the German physicist Gustav Kirchhoff formulated what are now known as Kirchhoff’s current law and Kirchhoff’s voltage law.

These laws are still widely used today, and in using them, more than one equation is usually set up. To find the needed information about the circuit, it is necessary to find solutions that satisfy all equations at the same time. We will show the solution of circuits using Kirchhoff’s laws in some of the exercises later in the chapter. Although best known for these laws of electric circuits, Kirchhoff is also credited in the study of optics as a founder of the modern chemical process known as spectrum analysis.

Methods of solving such systems of equations were well known to Kirchhoff, and this allowed the study of electricity to progress rapidly. In fact, 100 years earlier a book by the English mathematician Colin Maclaurin was published (2 years after his death) in which many well-organized topics in algebra were covered, including a general method of solving systems of equations. This method is now called Cramer’s rule (named for the Swiss mathematician Gabriel Cramer, who popularized it in a book he wrote in 1750). We will explain some of the methods of solution, including Cramer’s rule, later in the chapter.

Two or more equations that relate variables are found in many fields of science and technology. These include aeronautics, business, transportation, the analysis of forces on a structure, medical doses, and robotics, as well as electric circuits. These applications often require solutions that satisfy all equations simultaneously.

In this chapter, we restrict our attention to linear equations (variables occur only to the first power). We will consider systems of two equations with two unknowns and systems of three equations with three unknowns. Systems with other kinds of equations and systems with more unknowns are taken up later in the book.