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.
