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Conceptions About System of Linear Equations and Solution

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Challenges and Strategies in Teaching Linear Algebra

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Understanding of systems of linear equations permeates in the study of several topics of importance in linear algebra, such as rank, range, linear independence/dependence, linear transformations, characteristic values and vectors. After giving an overview of the literature on the teaching and learning of systems of linear equations, research results on student difficulties at different school and university levels are presented, establishing relationships with the way this topic is taught. The conceptions that students develop about ‘system’ and ‘solution’ are discussed in synthetic-geometric and analytic contexts in two and three dimensional spaces. Based on these observations, some pedagogical suggestions about planning instruction on this topic are offered. Although the findings reported in this chapter correspond to research undertaken in Mexico and Uruguay, they might be reflecting a more general phenomenon related to conceptions that students develop in relation with systems of linear equations and their solutions.

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Notes

  1. 1.

    Note that this question is repeated to see if the sequence of questions has made an effect on the students’ conceptions.

References

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Acknowledgements

I would like to thank Bonifacio Mora, Blanca Cutz, Cristina Ochoviet, Irving Alcocer, Carina Ramírez and Juan Guadarrama for their collaboration on parts of the project and data collection as well as the insights they brought to the project meetings.

Author information

Authors and Affiliations

  1. Cinvestav-IPN, Mexico City, Mexico

    Asuman Oktaç

Authors
  1. Asuman Oktaç
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Corresponding author

Correspondence to Asuman Oktaç .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, Oklahoma, USA

    Sepideh Stewart

  2. College of Education, Florida State University, Tallahassee, Florida, USA

    Christine Andrews-Larson

  3. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel

    Avi Berman

  4. Arizona State University, Mesa, Arizona, USA

    Michelle Zandieh

Appendices

Appendix 1: Questionnaire Applied to the First Group of University Students

  1. 1.
    1. (a)

      Draw the graphs of the following equations on the same coordinate system:

      \(4y = 3x - 5\) and \(y = \frac{6x - 10}{8}\)

    2. (b)

      Compare the graphs and write down your comments.

    3. (c)

      Use any method that you may know to solve the following system of equations:

      $$\left\{ {\begin{array}{*{20}c} {4y = 3x - 5} \\ {y = \frac{6x - 10}{8}} \\ \end{array} } \right.$$
  2. 2.
    1. (a)

      Draw the graphs of the following equations on the same coordinate system: \(y = \frac{2x + 3}{4}\) and \(12y = 6x + 10\)

    2. (b)

      Compare the graphs and write down your comments.

    3. (c)

      Use any method that you may know to solve the following system of equations:

    $$\left\{ {\begin{array}{*{20}c} {2x - 4y = - 3} \\ { - 6x + 12y = 10} \\ \end{array} } \right.$$
  3. 3.

    Solve the following system of equations:

    $$\left\{ {\begin{array}{*{20}c} {hx + y = 1} \\ {4x - 2y = k} \\ \end{array} } \right.$$
  4. 4.

    Given the following graphical representation of three lines, how many solutions does the system represented by these graphs have?

  5. 5.

    Solve the following system of equations:

    $$\left\{ {\begin{array}{*{20}c} {4x - 3y = - 1} \\ {2x + y = 4} \\ {x - 3y = 3} \\ \end{array} } \right.$$

Appendix 2: Questionnaire Applied to the Second Group of University Students

Part I. Question 1. Considering that in the first figure there are 2 lines in the plane and in all the other figures there are 3 lines, determine the number of solutions of the system of equations represented by each graph, what those solutions are and explain how you found them.

Question 2. Considering that in Fig. 7 there are 2 planes in the space and in all the other figures there are 3 planes, determine the number of solutions of the system of equations represented by each graph, what those solutions are and explain how you found them.

Part II. Question 3. For one of the previous figures write a system of equations that might represent it. (Although in the figures the coordinate axes were not included, you should have in mind the positions of the lines and planes with respect to each other).

Part III. Question 4. If it is possible, write a system of two equations in two unknowns so that it has:

  • (a) a unique solution        (b) no solution        (c) more than one solution

If it is not possible, explain why.

Question 5. If it is possible, write a system of three equations in three unknowns so that it has:

  • (a) a unique solution        (b) no solution        (c) more than one solution

If it is not possible, explain why.

Question 6. If it is possible, write a system of three equations in two unknowns so that it has:

  • (a) unique solution        (b) no solution        (c) more than one solution

If it is not possible, explain why.

Question 7. If it is possible, write a system of two equations in three unknowns so that it has:

  • (a) a unique solution        (b) no solution        (c) more than one solution

If it is not possible, explain why.

Part IV. Question 8. Solve the following system of equations:

$$\left\{ {\begin{array}{*{20}c} {4x - 3y = - 12} \\ {2x + y = 4} \\ {x - 3y = 3} \\ \end{array} } \right.$$

Appendix 3: Questionnaire Applied to the Middle and High School Students

  1. (1)

    The following figure shows lines associated to a system of three first degree equations in two unknowns. How many solutions does the system have? Why?

  1. (2)

    The following figure shows lines associated to a system of four first degree equations in two unknowns. How many solutions does the system have? Why?

  1. (3)

    The following figure shows lines associated to a system of three first degree equations in two unknowns. How many solutions does the system have? Why?

  1. (4)

    Solve the following system of equations using the method you wish. Does the system have a solution? If your response is negative explain why and if it is affirmative indicate how many solutions there are and what they are.

$$\begin{array}{*{20}c} {x + y = 2} \\ {x - y = 8} \\ {x + 2y = 4} \\ \end{array}$$
  1. (5)

    Can a system of three first degree equations in two unknowns have

    1. (a)

      a unique solution?

    2. (b)

      exactly two solutions?

    3. (c)

      and exactly three?

    4. (d)

      Can it have infinitely many solutions?

    5. (e)

      And no solution?

Explain each one of your answers and illustrate it by means of a graphical representation.

  1. (6)

    Can you put another line in the following figure so that the system of equations associated to all the lines has no solution? Explain your answer.

  1. (7)

    Can you put three more lines in the following figure so that the system of equations associated to all the lines has a unique solution? Explain your answer.

  1. (8)

    Can you put another line in the following figure so that the system of equations associated to two lines has only two solutions? Explain your answer.

  1. (9)

    Can you put another line in the following figure so that the system of equations associated to all the lines has infinitely many solutions? Explain your answer.

  1. (10)

    Can you put two more lines in the following figure so that the system of equations associated to all the lines has infinitely many solutions? Explain your answer.

  1. (11)

    Give a system of first degree equations whose unique solution is the ordered pair \(\left( {2,1} \right)\). Explain how you did it.

  2. (12)

    Can a system of first degree equations have as a solution the ordered pair \(\left( {2,1} \right)\) and also other solutions? If your answer is negative explain why it is not possible and if it is affirmative give an example explaining how you obtain it.

  3. (13)

    Can you put another line in the following figure so that the system of equations associated to them has the solution the ordered pair \(\left( {3,4} \right)\)?

  1. (14)

    Can you put another line in the following figure so that the system of equations associated to them has as its solutions only the ordered pairs \(\left( { - 3,2} \right)\) and \(\left( {2, - 1} \right)\)? Explain your answer.

  1. (15)

    Can you put another line in the following figure so that the system of equations associated to them has among its solutions the ordered pairs \(\left( { - 3,2} \right)\) and \(\left( {2, - 1} \right)\)? Explain your answer.

  1. (16)

    Can a system of three first degree equations in two unknowns have

    1. (f)

      a unique solution?

    2. (g)

      exactly two solutions?

    3. (h)

      and exactly three?

    4. (i)

      Can it have infinitely many solutions?

    5. (j)

      And no solution?

Explain each one of your answers and illustrate it by means of a graphical representation.Footnote 1

  1. (17)

    Explain what a system of equations is for you.

  2. (18)

    Explain what a solution of a system of equations is for you.

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Oktaç, A. (2018). Conceptions About System of Linear Equations and Solution. In: Stewart, S., Andrews-Larson, C., Berman, A., Zandieh, M. (eds) Challenges and Strategies in Teaching Linear Algebra. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-66811-6_4

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  • DOI: https://doi.org/10.1007/978-3-319-66811-6_4

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  • Online ISBN: 978-3-319-66811-6

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