# 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?

Question
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?

2021-03-05
Step 1
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 $$\displaystyle{A}={5}\times{6}$$ matrix.
dim(NulA)=1 because all solutions of
Ax = 0 are multiples of one nonzero solution.
Step 2
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 $$\displaystyle{5}\times{6}$$ matrix. n = 6
rank(A)=n-dim(NulA)=6-1=5
Image of A is 5-dimensional subspace of $$\displaystyle{R}^{{5}}$$(because A has 5 rows).
So, $$\displaystyle{C}{o}{l}{\left({A}\right)}={R}^{{5}}$$
This means that Ax=b has a solution for every b.

### Relevant Questions

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.
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$$\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}$$
$$\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}$$
$$\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=$$?