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 E_{1},
E_{2}, 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 laborhours 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.