# 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. Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss. Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent? Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent? In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix. Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions? When solving systems of equations we have at least two unknowns. A common example of a system of equations is a price problem. For example, Jacob has 60 coins consisting of quarters and dimes.
The coins combined value is \$9.45. Find out how many of each (quarters and dimes) Jacob has.
1.What do the unknowns in this system represent and what are the two equations that that need to be solved?
2.Finally, solve the system of equations.  $$\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}$$
$$\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}$$
$$\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=$$? 