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In algebra I, systems with two equations and two variables were solved (see systems. This chapter takes systems one step further and explains how to solve systems with three equations and three variables. This chapter will reveal three methods to solving such systems.

The first method is generalized from the Addition/Subtraction method. Systems with three equations and three variables can also be solved using the Addition/Subtraction method, and the first section details the steps involved in this process.

The second method involves the use of matrices and row reduction. Section two explains how to use row reduction to solve a system of three (or more) equations. This method can be thought of as a formalization of the Addition/Subtraction method. Both the first and the second method involve altering equations of the system without changing their meaning.

Like the second method, the third method uses matrices, but in a very different manner. It involves a quantity called the determinant, which is explained in the third section. Using the determinant and Cramer's Rule, you will be able to solve systems of equations without altering the equations. In fact, you will deal only with the coefficients and constants given in the original equations.

Systems of equations have applications in many fields--mathematics, physics, chemistry, biology, statistics--so it is useful to know how to solve them. This chapter presents three different methods for solving systems of equations. Although all three methods work for most systems, some are easier to use in certain situations. Thus, it is important to practice all three and become comfortable with their strengths and weaknesses.

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