3 Equations which have an infinite number of solutions

Consider the following system of equations

In augmented form we have:

$\left[\begin{array}{cccc}\hfill 1& \hfill 1& \hfill -3& \hfill 3\\ \hfill 2& \hfill -3& \hfill 4& \hfill -4\\ \hfill 1& \hfill -1& \hfill 1& \hfill -1\end{array}\right]$

Now performing the usual Gauss elimination operations we have

$\left[\begin{array}{cccc}\hfill 1& \hfill 1& \hfill -3& \hfill 3\\ \hfill 2& \hfill -3& \hfill 4& \hfill -4\\ \hfill 1& \hfill -1& \hfill 1& \hfill -1\end{array}\right]\begin{array}{c}\hfill \hfill \\ \hfill R2-2\times R1\hfill \\ \hfill R3-R1\hfill \end{array}\Rightarrow \left[\begin{array}{cccc}\hfill 1& \hfill 1& \hfill -3& \hfill 3\\ \hfill 0& \hfill -5& \hfill 10& \hfill -10\\ \hfill 0& \hfill -2& \hfill 4& \hfill -4\end{array}\right]$

Now applying $R2÷-5$ and $R3÷-2$ gives

$\left[\begin{array}{cccc}\hfill 1& \hfill 1& \hfill -3& \hfill 3\\ \hfill 0& \hfill 1& \hfill -2& \hfill 2\\ \hfill 0& \hfill 1& \hfill -2& \hfill 2\end{array}\right]$

Then $R2-R3$ gives

$\left[\begin{array}{cccc}\hfill 1& \hfill 1& \hfill -3& \hfill 3\\ \hfill 0& \hfill 1& \hfill -2& \hfill 2\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$

We see that all the elements in the last row are zero. This means that the variable $x_3$ can take any value whatsoever, so let $x_3=t$ then using back substitution the second row now implies

$x_2=2+2x_3=2+2t$

and then the first row implies

$x_1=3-x_2+3x_3=3-\left(2+2t\right)+3\left(t\right)=1+t$

In this example the system of equations has an infinite number of solutions:

$x_1=1+t,x_2=2+2t,x_3=t\text{or}{\left[x_1,x_2,x_3\right]}^T={\left[1+t,2+2t,t\right]}^T$

where $t$ can be assigned any value. For every value of $t$ these expressions for $x_1,x_2$ and $x_3$ 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.