Education Technology

Solving Systems of Linear Equations with Row Reductions to Echelon Form on Augmented Matrices

Activity Overview

This activity shows the user how to interpret a system of linear equations as an augmented matrix, row reduce the matrix to echelon form, and interpret the output to give a unique solution, generate infinite solutions, or conclude no solutions exist. The activity also shows how to check unique solutions. The activity examples can also be adapted to the TI-83+/84 family of calculators that contain rref() under the Matrix>Calc menu.

Before the Activity

All other methods of solving systems of equations should have been taught and tested prior to showing this calculator-based method. Using the rref command to reduce an augmented matrix to echelon form is a method that is easy to use when the solution to the system is more important than focusing on a particular skill set for solving the system. Students should know about putting linear equations in standard form. It would also be helpful to interpret a system of 2 equations with 2 unknowns as a 2x3 augmented matrix and to perform elementary row operations by hand to understand the transition from the input augmented matrix to the output matrix reduced to echelon form.

During the Activity

The students are shown in detail how to interpret a 3x3 system of linear equations as an augmented matrix, shown how to apply the rref() command to an input matrix using the TI-Nspire, and then how to interpret the Nspire's output with respect to stating the solution(s) to the system. The student will also learn how to check the unique solution to a system of linear equations. Students will be able to interpret output from row reduction to show a unique solution, generating equations that occur when the system has infinitely many solutions and an arbitrary variable, and also systems that have no solution. Systems are extended beyond those with n equations and n unknowns to others that have n equations and m unknowns.

After the Activity

Students should use this calculator-based method of solving systems of equations in the context of problem solving or working with nxm dimensional systems. One particular application is using row reduction to find the values of coefficients for a polynomial fit to a set of points as an alternative to data regression methods. The software called "Green Globs and Graphing Equations" contains the curve/function fitting game Green Globs that puts 13 large globs on a coordinate plane. The object of the game is to define a function that goes through as many globs as possible. The rref() method of solving a system can be used to solve for the coefficients of a polynomial function that goes through selected points on the graph. Selecting 3 points is a basis for a 3x3 systems to find the coefficients of a quadratic function. Selecting 4 points allows the student to find coefficients for a cubic polynomial. The score in this game grows exponentially according to S(x)= 2^n-1 for hitting n globs with a single function, so there is an incentive to select points that will hit other other globs along the path of the polynomial in addition to the selected globs.