Solving Systems of Linear Equations with the Help. of Free Technology

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1 Solving Systems of Linear Equations with the Help of Free Technology Calin Galeriu, Ph.D. 1. Introduction The use of computer technology when teaching new math concepts, or when solving difficult math problems, is a well established and recommended practice. Student friendly technology can decrease math anxiety in the classroom, improve student understanding of abstract algebra topics, simplify math calculations, and lead to more motivated students. Students with diverse learning styles need to see a variety of mathematical approaches, and technology often delivers that easier, more visual, or more powerful method. Aspiring math teachers are now expected to learn how to use educational technology, since technology provides such an effective approach to mathematical exploration and intuitive learning. Some advanced problems cannot be solved with only pencil and paper. The use of a programming language like BASIC becomes necessary when students explore approximate numerical solutions, convergence of series, and iterative or Monte Carlo methods. Computer programming in BASIC is an activity that can integrate technology into many mathematics courses. Computer programming is one excellent way to promote abstract reasoning and critical thinking. It encourages students to put structure in their thoughts, organize data clearly, think sequentially, internalize procedures and algorithms, and imagine new ways of solving problems. The best results are achieved when students have uninterrupted access to technology, at school or at home, allowing for around the clock practice. With this idea in mind we present here three excellent free computer applications (GeoGebra, Eigenmath, and a BASIC 1

2 interpreter) that can be used for mathematical activities. Because students are able to download and install the software at home they can get a better sense of ownership of the math tools provided. The three programs described complement each other, as exemplified by the following discussion about systems of linear equations. 2. GeoGebra GeoGebra is a software application that allows a hands-on approach to geometry and algebra, in an integrated environment. A GeoGebra object, usually a function, can have three representations: an algebraic definition, a visual graph, and a table of input and output values. Changes to one of the representations will immediately affect the other representations. This provides an interactive way to explore the connection between mathematical concepts. A manual is available online on the GeoGebra website ( As an example, we will solve the following system of equations by graphing: 2 x + 3 y = 4, 5 x + 6 y = 7. Using GeoGebra we can graph the two lines, we can find the intersection point, and we can identify the x and y coordinates of that point. In the Input Bar we have to type: 2x+3y=4<ENTER>, 5x+6y=7<ENTER>, Intersect[a,b]<ENTER>. 2

Figure 1. Solving a system of linear equations by graphing. While not the most powerful method, solving a system of linear equations by graphing is visual, very intuitive, and very easy to understand.

3 Figure 1. Solving a system of linear equations by graphing. While not the most powerful method, solving a system of linear equations by graphing is visual, very intuitive, and very easy to understand. The graphical method also provides the best way to explain the three possible answer outcomes: one solution (intersecting lines, one intersection point), no solution (parallel lines, no intersection), and infinitely many solutions (overlapping lines). 3. Eigenmath The program Eigenmath, written by George Weigt, rivals a graphing calculator. This powerful computer algebra system can, among other things, manipulate mathematical expressions in symbolic form, work with matrices, solve systems of linear equations, make graphs of parametric two-dimensional functions, and perform differentiation and integration. 3

The newest version of Eigenmath is called EVAlgebra, and can be downloaded from http://sourceforge.net/projects/evalgebra/. We will solve our system of equations with the help of the inverse matrix.

4 The newest version of Eigenmath is called EVAlgebra, and can be downloaded from We will solve our system of equations with the help of the inverse matrix. First we have to write the system of linear equations in matrix form (A X = B): x 4 =. 6 y 7 If the square matrix of the coefficients has an inverse, then the solution is obtained by multiplying both sides of the equation by the inverse matrix (X = A 1 B). In Eigenmath this is done by typing: A=((2,3),(5,6))<ENTER>, B=(4,7)<ENTER>, X=dot(inv(A),B)<ENTER>, X<ENTER>. One could also type just dot(inv(a),b)<enter>. Figure 2. Solving a system of linear equations with the help of the inverse matrix. Solving a system of two linear equations with the help of the inverse matrix is a very fast procedure. The method also works very well for systems with more than two equations. 4

5 The shortcoming of this method is that, most often, high school students don t have a clear understanding of the steps involved in finding the inverse of a matrix, and they have to rely exclusively on the computer for this task. 4. A BASIC Interpreter While computer programming in BASIC has a longer learning curve, the results are often worth the effort. Today s computers don t come with a simple BASIC compiler already installed, like it used to be. An option for students is to use the free online BASIC interpreter written by Joshua Bell and posted online at The BASIC program listed in Figure 3 solves a system of two equations of the form: a x + b y = c, d x + e y = f. The solution formulas c e b f x =, a e b d a f c d y =, a e b d are easily derived by solving the general system of two equations by linear combinations, performing symbolic algebraic manipulations from start to finish. 5

6 Figure 3. Solving a system of two linear equations using a BASIC program. This method promotes the use of variables to its fullest extent, which is one of the mathematical habits of mind that we should cultivate in our students. Often in algebra one wants to delay numerical evaluations until the end of a process so that one can see how the operations are sequenced and so that the structure of the calculation (rather than its value) becomes more apparent [1]. The flowchart of a BASIC program can often provide a new perspective, a deeper insight, or a better explanation of a mathematical procedure or algorithm. 5. Discussion of the Different Methods of Solving Systems of Linear Equations An engaging class discussion will follow the presentation of these different solution methods. This discussion is of fundamental importance, and making connections between the different 6

7 methods is as important as learning how to solve a system of two linear equations by any one of the methods. Assuming that the coefficients b and e are not equal to zero, the two equations can be written in slope-intercept form: a c y = x +, b b d f y = x +. e e The first line has a slope of second line has a slope of a c c, a y-intercept of, and an x-intercept of. The b b a d f f, a y-intercept of, and an x-intercept of. e e d The two lines are parallel (and the system of equations has no solution) when the two slopes are the same. This condition is equivalent to a e b d = 0, the equation tested in line 200. Even more, when the two lines are parallel and vertical ( b = 0 and e = 0 ) the equation a e b d = 0 is still satisfied. The condition a e b d = 0 also implies that the square matrix of the coefficients has a zero determinant, and as a result there is no inverse matrix. The two lines overlap (and the system of equations has infinitely many solutions) when the two slopes are the same and the two y-intercepts are the same. The second condition is equivalent to c e b f = 0, and this is the first equation tested in line 300. In the special situation when both lines are vertical we have to compare the two x-intercepts. The two x-intercepts are the same when a f c d = 0, and this is the second equation tested in line 300. Furthermore, by writing the solution formulas in matrix form e x a e b d = y d a e b d b a e b d c, a f a e b d 7

8 and by comparing this equation to X = A 1 B, we can also identify the formula for the inverse matrix. The inverse of the two-by-two matrix a d b e 1 e is a e b d d b a, provided that a e b d 0. Additional insight into the diverse nature of the students learning styles can be revealed by asking them: What is your favorite method of solving systems of two linear equations, and why? 6. References [1] A. Cuoco, E. P. Goldenberg, and J. Mark, Organizing a Curriculum around Mathematical Habits of Mind, Mathematics Teacher, 103 (2010), Common Core Standards Addressed by This Unit A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimensions 3 x 3 or greater). Copyright Calin Galeriu, All rights reserved. This document can be downloaded for free at 8

Common Core State Standards for Mathematics

Common Core State Standards for Mathematics A Correlation of Pearson to the for Mathematics for Mathematics Introduction This document demonstrates how Pearson s digits program meets the for Mathematics. Correlation references are to the digits