Systems of Linear Equations in Three Variables and Matrix Operations

Systems in Three Variables
Systems of the form

are called linear equations in three variables. A triplet of numbers (x,y,z) is a solution of the system if EACH equation is satisfied by the triplet. Figure 1 (p. 483) shows several of the many ways in which three planes can intersect:

Using Elimination by Addition
The general approach to solving a system with three variables is to use Theorem 2 in the last section to eliminate variables until an equivalent system with an obvious solution is obtained. We will refer to the equations in a system as E₁, E₂, and so on.

Example 1: Solve:

Performing Row Operations on Matrices

A matrix is a rectangular array of numbers written within brackets. Each number in a matrix is called an element of the matrix. If a matrix has m rows and n columns, it is called an m×n matrix. The expression m×n is called the size of the matrix, and the numbers m and n are called the dimensions of the matrix.

*Note:

If the number of rows is equal to the number of columns, then the matrix is called a square matrix. A matrix with only one column is called a column matrix, and a matrix with only one row is called a row matrix.

Ex.

The position of an element in a matrix is the row and column containing the element. This is usually denoted using double subscript notation a_ij,
where i is the row and j is the column containing the element a_ij.

Ex.

The principal diagonal of a matrix consists of the elements a_ii, i=1,2,…,n.
Example 1: Write the augmented coefficient matrix corresponding to each of the following systems.

Recognizing Reduced Matrices
Example 3: Write the system corresponding to each of the following augmented coefficient matrices and find its solution.

*Note:

Go back and try example 1 using matrices.
Recognizing Inconsistent and Dependent Systems

As with systems with two variables and two equations, if you obtain an inconsistent equation (such as 0=1) while solving a system, then the system is inconsistent and has no solution. On the other hand, if you obtain an equation that is always true (such as 0=0), then the system may be inconsistent (no solution) OR dependent (infinite number of solutions). The solution process must proceed further to determine which is the case.

Example2: Solve:

Example 3: Solve:

Modeling with Systems in Three Variables
Example 4: A garment industry manufactures three shirt styles. Each style shirt requires the services of three departments as listed in the table. The cutting, sewing, and packaging departments have available a maximum of 1,160, 1,560, and 480 labor-hours per week, respectively. How many of each style shirt must be produced each week for the plant to operate at full capacity?

	Style A	Style B	Style C	Time available
Cutting department	0.2 hr	0.4 hr	0.3 hr	1,160 hr
Sewing department	0.3 hr	0.5 hr	0.4 hr	1,560 hr
Packaging department	0.1 hr	0.2 hr	0.1 hr	480 hr

Example 5: In 2001 there were 110 million cell phone subscribers in the United States. This number grew to 128 million in 2002 and 141 million in 2003. Construct a model for these data by finding a quadratic function whose graph passes through the points (1,110), (2,128), and (3,141). Use this model to estimate the number of subscribers in 2004 and 2005.