Folio: Investigating Systems of Linear Equations: a) General Case for a 2×2 System; b) Mathematical Modelling. Purpose To demonstrate your ability to: • Use mathematics as a tool to analyze data and other information elicited from the study of situations taken from different contexts • Use electronic technology appropriately • Work cooperatively in planning and carrying out mathematical investigation • Communicate relevant information in the form of a written report. Description of assessment This assessment allows you to show your skills in understanding and appropriate use of the mathematical concepts, processes, and strategies in the following: • Subtopic 3.1: Using Linear Equations • Subtopic 3.2: Solutions to Systems of Equations Assessment conditions where students demonstrate their progress in the form of notes, drafts, calculations, etc. Learning Requirements Assessment Design Criteria Capabilities ” 1. Understand fundamental mathematical concepts, demonstrate mathematical skills and apply routine mathematical procedures 2. Use mathematics as a tool to analyze data and other information elicited from the study of situations taken from social, scientific, economic, or historical contexts 3. Think mathematically by posing questions/problems, making and testing conjectures, and looking for reasons that explain the results 4. Make informed and critical use of electronic technology to provide numerical results and graphical representations 5. Communicate mathematically and present mathematical information in a variety of ways 6. Work both individually and cooperatively in planning, organizing, and carrying out mathematical activities. ” “Excerpt from Mathematical Studies 2013 Subject Outline, page 47: A completed investigation should include: • an introduction that outlines the problem to be explored, including its significance, its features, and the context • the method required to find a solution, in terms of the mathematical model or strategy to be used • the appropriate application of the mathematical model or strategy, including – the generation or collection of relevant data and/or information, with details of the process of collection – mathematical calculations and results, and appropriate representations – the analysis and interpretation of results – reference to the limitations of the original problem • a statement of the results and conclusions in the context of the original problem • appendices and a bibliography, as appropriate. The investigation consists of two parts: Part A: Investigation a 2×2 system (General Case) (Individual work, guided investigation) Part B: Application of Systems of Linear Equations in Mathematical Modeling (Open-ended Task) On the basis of the above, the following report structure is recommended: 1. Introduction 2. Mathematical Investigation and Analysis Part A: a) Task 1 b) Task 2 c) Task 3 Part B: a) Modeling of a situation b) Description of group work and personal contribution 3. Conclusion 4. Appendix (optional) 5. Bibliography Part A: Investigation a 2×2 system (General Case) (Individual work, guided investigation) Theoretical Background In this investigation you have to provide a complete and detailed analysis of a system of 2 linear equations in 2 unknowns with arbitrary values of the parameters (coefficients). The necessary definitions: Definition 1 A system of two linear equations in two unknowns x and y is a system of two equations of the form: where are arbitrary real numbers.
 Definition 2 The solution set of the system is a collection of pairs of real numbers (x0, y0) such that each pair (x0, y0) satisfies each of the equations of the system. To solve a system of equations means to find all the solutions (i.e. the entire solution set). Definition 3 A system of equations is said to be: • Consistent if it has at least one solution; • Inconsistent if it has no solution; A consistent system is said to be: • Determinate if it has a unique solution, i.e. only one pair (x0, y0) in the solution set; • Indeterminate if it has more than one solution in the solution set. Geometrical Interpretation: We know that the set of points in a plane whose coordinates (x, y) satisfy an equations of the form , where either a or b (or both) are nonzero, constitutes a straight line. In other words, the geometrical image (interpretation) of the equation is a straight line. If some values of the coefficients a, b, or c are zeros, we have two special cases: • If , then all pairs of numbers (x, y) satisfy the equation. This means that for these values the equation constitutes the entire xy-plane. In other words, the geometrical image (interpretation) of the equation is the entire xy-plane. • If , but : no pair (x, y) satisfies the equation. In this case we say that the image (interpretation) of the equation is an empty set. It is clear that in the case of 2 linear equations, i.e. the system (1), we may have different combinations of lines, planes or empty sets. For example, it may be: • Line and line (which may intersect, or may be parallel or may coincide) • Line and empty set • Plane and line • etc. * * * Your objective is to investigate system (1). This means to decide whether it is consistent or inconsistent, and if the system is consistent to decide whether it is determinate or indeterminate. In the two latter instances, it is necessary to find the solutions. You will also have to provide a geometrical interpretation of the system and its solution(s). Task 1 Examples of Systems Considering several examples of the 2×2 systems, decide whether the system is consistent, inconsistent, determinate or indeterminate. Explain your decision in each case. If a system has solution(s), you may also find them. Task 2 Investigation of the General Solution of System (1) Consider the general case of system (1). 1. Write down the augmented matrix of the system and using row operations, transform the matrix to the reduced echelon form: (2) where is the principal determinant of the system (1); is the x-determinant of the system (1); is the y-determinant of the system (1). 2. Using the definition of the determinate of a matrix from Unit G, page 386, Mathematical Studies, Year 12, write down in the form of a 2×2 array. 3. Investigate the case when . Write down the solutions in terms of The solution written in terms of constitutes the Cramer’s rule. Formulate the Cramer’s rule. 4. Investigate the case when . This case contains 4 sub-cases: a) or (i.e. at least one of the determinants , is not zero). b) but at least one of the coefficients is not zero. For the sake of definiteness, let , whereas the values of may or may not be zeros. c) and but (at least one of the numbers is not zero). d) and and 5. Provide a geometrical interpretation of the system (1) and its solution(s) for all cases investigated above. Task 3 Application Application 1 Consider a 2×2 system: a) Show that the system has a unique solution, and solve it using the Cramer’s rule; b) Solve the system by using row operations. For this, reduce the augmented matrix of the system to the reduced echelon form and find the solutions. c) Check that the both methods produce the same result. Application 2 Consider a 2×2 system: Solve the system providing a complete investigation of all possible cases. Part B: Application of Systems of Linear Equations in Mathematical Modelling (Open-ended Task) In this task you have to complete the following: Setting a Mathematical model and Applying it to Solve and Interpret a process (phenomenon or situation) • Via research, suggest a process, phenomenon or situation in physical, chemical, biological, economic, social, or other context • Propose a general model for the suggested process, phenomenon, or situation • By specifying some parameters of the process (phenomenon or situation), derive a system of linear equations • Provide a solution of the system to specify the model. • Provide analysis and interpretation of the model and results • Discuss possible limitations of: (a) the model, (b) of the obtained solution (results) • Discuss the assumptions made. page – 386 Unit G
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