Solving Systems of Linear Equations With Row Reductions to Echelon Form On Augmented Matrices. Paul A. Trogdon Cary High School Cary, North Carolina

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1 Solving Sstems of Linear Equations With Ro Reductions to Echelon Form On Augmented Matrices Paul A. Trogdon Car High School Car, North Carolina There is no more efficient a to solve a sstem of linear equations than ith Ro Reduction to Echelon Form (rref) on an augmented matri. There are a variet of algebraic, geometric, and numerical methods available, but using the rref command applied to an input augmented matri handles all possible cases of solving sstems of linear equations. The parts of the folloing activit ill sho ho to use the TI-Nspire to deal ith a variet of sstems, some of hich have unique solutions, infinitel man solutions, or no solutions at all. (NOTE: The TI-8/8 also have the RREF command at MATRIX > CALC > RREF to use on an input augmented matri.) To get started, look at the eample of a sstem of linear equations containing variables/unknons. GIVEN the sstem of linear equations in standard form: 7 The coefficients and constants in the sstem are going to be entered as a b matri, one ro for each equation. The folloing screen shots and instructions ere captured on a TI-Nspire CAS calculator, but the same kestrokes ill ork similarl on the regular TI-Nspire calculator as ell. On the TI-Nspire CAS, open a document : Add Calculator page Choose b 8: Matri & Vector (NOTE: This is option 7 on the Nspire.)

2 : Reduced Ro-Echelon Form to create an rref() command. With the cursor inside the parenthesis,. Choose /r Matri

3 Select ros and e to enter columns, then e to OK to create the template to receive the values for this sstem The field of ros and columns are set to receive the coefficients and constants from the sstem in the eample. NOTE: It is best to use the e ke to move from one entr field to the net. The e ke ill move through the fields column b column and ro b ro. The arro kes on the NAVPAD are not as automatic as the e ke for navigating the field.

4 Use the is after value entr is complete. The matri ill automaticall be ro reduced to echelon form shon on the right as output. The output in the last screen shot translates into the folloing set of equations revealing the solutions to the sstem of linear equations for this eample: or It should be eas to see that the ro reduced form gives the solution for the values of,, and in the last column. Reduction to eschelon form indicates that the main diagonal is populated ith s and elements off the diagonal are s hen the sstem has a unique solution. Hoever, to other possibilities can occur hen RREF is applied to an input augmented matri representing some sstem of equations. The first alternative is a sstem that has no solutions. A reduced-echelon form that has a ro of eros in the coefficient section and a non-ero number in the augmentation column occurs ith a sstem that has no solution. The folloing is an eample of a sstem translated into an augmented matri and the outcome that occurs hen RREF is applied to the sstem. Follo the steps above for entering this eample. Remember to use the e ke for moving through the entr fields of the matri ou created. 8 should be entered as 8 rref, then press. The output should be 7. The last ro translates into the false statement, indicating no solution.

5 The second alternative is a sstem ith infinitel man solutions. The folloing is an eample of a sstem translated into an augmented matri and the outcome that occurs hen RREF is applied to the sstem: should be entered as rref, then press The output should be. The last ro translates to the true statement indicating infinitel man solutions. In this second case, the variable is considered a value open to selection or an arbitrar variable. Hoever, the values of and var along ith the choice of value for variable. Consider the analog to the social process of one person becoming engaged to marr another person. With the choice of future spouse also comes the selection of future mother- and father in-la and a hole bunch of other in-las, but the initial selection of a significant other is a more-or-less arbitrar choice. The same is true ith choice of and the subsequent determination of the values of and. To generate members of the solution set, consider the ro-reduced form of the previous eample: translates back into the sstem is arbitrar or is arbitrar Here are some of the infinitel man solutions that eist for this sstem based on random choices of. All of the solutions are generated b the equations for and : (,,),,,, - ( ) (,,) -,,

6 EXAMPLES OF OTHER SYSTEMS OF EQUATIONS: (NOTE: Refer back to the calculator entr instructions on the first page of this activit.) Eample #: A sstem of to equations ith to unknons: On the TI-Nspire: rref. Output should be, indicating the solutions are and. Eample #: Arrange the sstem of to equations ith to unknons into standard form (or solve the sstem b graphing the to lines to find their point of intersection!). restates as. On the TI-Nspire: Output should be rref., so the solutions are and. Eample #: An etra equation is included in the sstem ith to unknons to be identified. 7 On the TI-Nspire solve an augmented matri choosing the first to of the equations given. This overstocked sstem has solution (,). Remember that a solution to a sstem must make all equations in the sstem true, so check the solution from the to equations selected in the third equation. If the ordered pair fails in the rd equation, the sstem has no solution.

7 Eample #: A sstem of equations ith unknons, so our augmented matri ill have ros and columns. 8 7 Solve ith TI-Nspire: 8 7 rref. Output should be, so the solutions are ( ),,,,,,. This is a good time to use the TI-Nspire to check the solution to the sstem numericall. The/h ke is used to assign a numerical value to a variable from the sstem. The values can be stored one per entr line or altogether as shon ith the STO commands separated b colons. The left member of each equation is entered as it appears in the sstem folloed b an ke. The value of the equation should agree ith the constant for that equation in the sstem. You can also enter the entire equation and check for a true or false classification on the entr. This method ould be particularl easier is cases here variables are present in both members of the equations.

8 Eample #: This sstem has onl equations, but variables, so the sstem could have infinitel man solutions and an arbitrar variable or no solution at all. 8 On the TI-Nspire: rref. 8 The output should be Notice that the output shos the matri is reduced to echelon form through the first three columns. This is a prett mess output in this eample, but it contains the equations that generate all solutions to the sstem based on the arbitrar selection of a value for the variable. restated to generate values of,, and using s value: is arbitrar SUMMARY: Ro reduction operations on an augmented matri are an effective method for solving an n X m sstem of linear equations using the Nspire s rref command. The output is easil interpreted to give unique solutions, generate solutions based on an arbitrar variable, or indicate the sstem has no solution. When a correct solution is the goal rather than shoing a particular method, ro reducing the sstem to echelon form is the shortest path to that goal.