Consider the provided question,

Given a system of five linear equations in six unknowns are all multiples of one nonzero solution.

It can be written as,

\(Ax=0\) where \(A=5\times6\) matrix

\(dim(NulA)=1\) because all solutions of \(Ax=0\) are multiples of nonzero solution

We have to explain that the system necessarily have a solution for every possible choice of constants on the right sides of the equations.

Since,A is \(5\times6\) matrix \(n=6\)

\(rank(A)=n-dim(NulA)=6-1=5\)

Image of A is 5 dimensional subspace of \(R^5\) (because A has 5 rows)

So,\(Col(A)=\)\(R^5\)

This means that Ax=b has a solution for every b