3 Equations which have an infinite number of solutions
Consider the following system of equations
In augmented form we have:
Now performing the usual Gauss elimination operations we have
Now applying and gives
Then gives
We see that all the elements in the last row are zero. This means that the variable can take any value whatsoever, so let then using back substitution the second row now implies
and then the first row implies
In this example the system of equations has an infinite number of solutions:
where can be assigned any value. For every value of these expressions for and will simultaneously satisfy each of the three given equations.
Systems of linear equations arise in the modelling of electrical circuits or networks. By breaking down a complicated system into simple loops, Kirchhoff’s law can be applied. This leads to a set of linear equations in the unknown quantities (usually currents) which can easily be solved by one of the methods described in this Workbook.