# Systems of Equations

## Systems In Three Or More Variables

Systems of equations with more than two variables are possible. A linear equation in three variables could be represented by the equation ax + by + cz = k, where a, b, c, and k are constants and x, y, and z are variables. For these systems, the solution set would contain all the number triplets which make the equation true. To obtain the solution to any system of equations, the number of unknowns must be equal to the number of equations available. Thus, to solve a system in three variables, there must exist three different equations which relate the unknowns.

The methods for solving a system of equations in three variables is analogous to the methods used to solve a two variable system and include graphical, substitution, and elimination. It should be noted that the graphs of these systems are represented by geometric planes instead of lines. The solutions by substitution and elimination, though more complex, are similar to the two variable system counterparts.

For systems of equations with more than three equations and three unknowns, the methods of graphing and substitution are not practical for determining a solution. Solutions for these types of systems are determined by using a mathematical invention known as a matrix. A matrix is represented by a rectangle array of numbers written in brackets. Each number in a matrix is known as an element. Matrices are categorized by their number of rows and columns.

By letting the elements in a matrix represent the constants in a system of equation, values for the variables which solve the equations can be obtained.

Systems of equations have played an important part in the development of business, industry and the military since the time of World War II. In these fields, solutions for systems of equations are obtained using computers and a method of maximizing parameters of the system called linear programming.

## Resources

### Books

Barnett, Raymond, and Michael Ziegler. College Mathematics. San Francisco: Dellen Publishing Co, 1984.

Bittinger, Marvin L. and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Paulos, John Allen. Beyond Numeracy. New York: Alfred A. Knopf Inc., 1991.

Perry Romanowski

## KEY TERMS

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Consistent system

—A set of equations whose solution set is represented by only one ordered pair.

Dependent system

—A set of equations whose solution set has an infinite amount of ordered pairs.

Elimination

—A method for solving systems of equations which involves combining equations and reducing them to a simpler form.

Graphical solution

—A method for finding the solution to a system of equations which involves graphing the equations and determining the points of intersection.

Inconsistent system

—A set of equations which does not have a solution.

Linear equation

—An algebraic expression which relates two variables and whose graph is a line.

Matrix

—A rectangular array of numbers written in brackets and used to find solutions for complex systems of equations.

Ordered pair

—A pair of values which can represent variables in a system of equations.

Solution set

—The set of all ordered pairs which make a system of equations true.

Substitution

—A method of determining the solutions to a system of equation which involves defining one variable in terms of another and substituting it into one of the equations.