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Systems of Linear Equations in Three Variables and Matrix OperationsSystems in Three Variables
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 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}, Ex. The principal diagonal of a matrix consists of the
elements a_{ii}, i=1,2,…,n. Recognizing Reduced Matrices *Note: Go back and try example 1 using matrices. 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 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.
