It looks to me like they are just numbered wrong. That is "system 5" refers to the two equations on the left, 5x+ y= 3 and 2x- y=4, while "system 6" refers to the two equations on the right $\displaystyle 4x_1- x_2+ 3x_3= -1$ and $\displaystyle 3x_1+ x_2+ 9x_3= -4$.
The first system, two equations in two unknowns, has the unique solution x= 1, y= -2: 5(1)- 2= 3, 2(1)-(-2)= 4.
The second system, two equations in three unknowns, has an infinite number of solutions.