03 - Linear System - Example 3 - Solving Linear Systems (No Solutions)

Determine the solutions (if any) to the following equation

$$x_1+x_2+x_3+x_4=4;2x_1+3x_2-2x_3-3x_4=1;x_1+5x_3+6x_4=1$$

Solution (EROs)

There be less detail shown for the exact EROs but it is identical in concept to Linear System - Example 1

Applying EROs again

$$\left(\begin{array}{ccccc}1& 1& 1& 1& 4\\ 2& 3& -2& -3& 1\\ 1& 0& 5& 6& 1\end{array}\right)\to \left(\begin{array}{ccccc}1& 0& -5& -6& -11\\ 0& 1& -4& -5& -7\\ 0& 0& 0& 0& 1\end{array}\right)$$

No solution as converting back to the Linear System form (using row $3$) gives

$$0x_1+0x_2+0x_3+0x_4=1$$

Which has no solutions