# Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

Forms of linear equations
Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

2020-12-25

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