Chapter 24: Simultaneous Equations: Situation 18 From the MACMTL-CPTM Situations Project

A student teacher in a course entitled Advanced Algebra/Trigonometry presented several examples of solving systems of three linear equations in three unknowns algebraically using the method of elimination (linear combinations). She started another example and had written the following

When a student asked, “What if you only have two equations?”

Knowing necessary and sufficient conditions for unique solutions to systems of linear equations is important in this Situation. The Foci build from systems of equations in two variables to systems of equations in three variables, and examine why n independent equations are necessary to produce a unique solution to a system of equations in n variables. Systems of linear equations in two or three variables may be consistent or inconsistent and dependent or independent. If a set of equations has no solutions in common, the system is referred to as being inconsistent. If a set of equations has at least one common solution, the system is referred to as being consistent. When a consistent system has exactly one solution, it is referred to as being independent ; if it has more than one solution, it is referred to as being dependent.¹ The Foci use physical models, symbolic representations, graphical representations, and matrix representations to examine systems of linear equations with unique solutions, an in?nite number of solutions, and no solutions.