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# HP 49G Series and 48GII Calculators - Solving Linear Systems of Equations with Variables in the Coefficients

Introduction
Matrices with variables need to be resolved using the RREF command on the calculator.
The following system of equations will be used for the example in this document:
Ax + 5y = 10
5x + y = B
Normally, a system of linear equations is solved by using the "Solve Linear System" option under the numeric solver. However, attempting to enter the variable A for the first coefficient will cause a "Bad Argument Type" error to be displayed. This is because the linear system solver is designed to solve problems numerically, not symbolically. The solution to this problem is to use the RREF command instead of the linear solver. RREF stands for "reduced row echelon form," which requires the first nonzero entry in each row to be a one and also the only nonzero entry in its column.
Entering the matrix
The first step in solving a problem in this manner is to enter a single matrix representing the entire linear system. This is the same step that would be done when solving the equation using linear algebra with an augmented matrix.
##### note:
Make sure that both the "numeric" and "approximate" options are not checked in the CAS before entering the matrix.
For the example above, in RPN mode, the matrix would look like the following (see Figure 1):
Figure :
In algebraic mode, the matrix would look like the following (see Figure 2):
Figure :
Using the RREF command
After placing the matrix on the stack, execute the RREF command.
Either type the command in with the alpha keys, or follow these steps:
1. Press left-shift, then MATRICES.
2. Select LIN S (linear systems).
3. Select RREF.
In algebraic mode, the syntax would be RREF([[A 5 10] [5 1 B]]).
The calculator will return the following matrix (see Figure 3):
Figure :
The left portion of this matrix is the identity. The right column is the solution vector.
There is more information on matrix manipulation in Chapter 5 of the Advanced Users Guide, which can be downloaded at the following URL:
http://www.hp.com/calculators/techsupport/graphing/49g_userguide/

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