The augmented matrix of a homogeneous system with four variables and five equations has rank 4. What can you say about the solutions of this system?
Label the four variables `x_1,x_2,x_3,x_4.` Consider the reduced row echelon form of the augmented matrix. Because the system is homogeneous, the rightmost column consists of all zeros. But there are five columns and the rank is four, so each of the first four columns must have a 1 in a pivot position. Since all rows of zeros are at the bottom, the RREF is
which says that `x_1=x_2=x_3=x_4=0.`
The system has only the trivial solution.