Until now, we have been working with single equations. We have seen equations with one variable, which generally have a finite number of solutions, and equations with two variables, which are usually satisfied by an infinite number of ordered pairs. In this section, we will begin to deal with systems of equations; that is, with a set of two or more equations with the same variables. We limit our discussion to systems of linear equations, since our techniques for solving even a single equation of higher degree are quite limited.
Systems of linear equations can have zero, one, or an infinite number of solutions, depending on whether they are consistent or inconsistent, and whether they are dependent or independent. The first section will explain these classifications and show how to solve systems of linear equations by graphing.
The second section will introduce a second method for solving systems of linear equations--substitution. Substitution is useful when one variable in an equation of the system has a coefficient of 1 or a coefficient that easily divides the equation.
If one of the variables has a coefficient of 1, substitution is very useful and easy to do. However, many systems of linear equations are not quite so neat, and substitution can be difficult. The third section introduces another method for solving systems of linear equations--the Addition/Subtraction method.
Systems of equations will reappear frequently in Algebra II. They will be used in maximization and minimization problems, where solving by graphing will become especially useful. Systems of equations also appear in chemistry and physics; in fact, they are found in any situation dealing with multiple variables and multiple constraints on them.