Solving systems of equations
Nodal analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters. Nodal analysis creates linear equations using Kirchhoff's Laws of junctions and paths. This is a popular project for students who have studied some physics. One value of this project is the ability to create overdetermined, consistent systems of equations, which helps students understand rows of zeros in the RREF form of augmented matrices. This article on Nodal Analysis of Electric Circuits has a clear explanation.
Loop analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters. Loop analysis creates linear equations using Kirchhoff's Laws of loops. Again, a popular project for students who have studied some physics and also has the opportunity for overdetermined, consistent systems. Equivalent in results to nodal analysis, this could be combined or assigned separately. This article on Loop Analysis of Electric Circuits has a clear explanation.
Curve fitting - Using systems of equations a student finds the coefficients of a polynomial of degree n - 1 to fit n points. I don't think of this as a juicy application that gives the student an appreciation for how linear algebra is used in the world. Fitting an n-degree polynomial to m points using least squares or other methods is more likely to happen.
