# Non homogeneous system question

• nhrock3
In summary, the conversation discusses a system Ax=b where x and b are vectors, and A is a system of kxn type. If k>n, the system has endless solutions due to free variables. If k=n and Ax=b has a single solution, then for any system Ax=c has a solution, which can be found by performing the same row operations on c as were performed on b to find the solution for x.
nhrock3
there is a system Ax=b (x and b are vectors)
A is a system of kxn type
the system has at leas one solution:
1.
if k>n does the system has endless solutions
??
2.
if k=n and Ax=b has a single solution then for any system Ax=c has a solution
??

for 1:
i don't know why we need the other data k<n means we have free variables so its endless solutions

for 2:
i don't know what's c

For 1, you are answering a different question than the one given.

For 2, if you are given that k = n and Ax = b has a unique solution, how would you find this solution? How would you find the solution to Ax = c?

Last edited:
one of the ways i think about part 2 is that if Ax=b has a unique solution, then if we take the matrix A and augment it with b, then A can row reduce down to a nxn identity matrix. the augmented part depends on b, certainly, but if A reduces down, then b doesn't really matter, at least considering the row operations required to get the identity. So if we keep track of the row operations we performed on A to get the solution for b, we can perform those same row operations on just any vector c to get the appropriate solution.

## 1. What is a non-homogeneous system?

A non-homogeneous system is a system of linear equations where not all of the equations have a constant term of zero. This means that there is at least one equation in the system that has a non-zero constant term.

## 2. How is a non-homogeneous system different from a homogeneous system?

A homogeneous system is a system of linear equations where all of the equations have a constant term of zero. This means that the system has only the trivial solution, where all variables are equal to zero. A non-homogeneous system, on the other hand, can have non-zero solutions.

## 3. How can you solve a non-homogeneous system?

To solve a non-homogeneous system, you can use methods such as Gaussian elimination or substitution. These methods involve manipulating the equations in the system to reduce it to a simpler form and then solving for the variables.

## 4. Can a non-homogeneous system have more than one solution?

Yes, a non-homogeneous system can have infinitely many solutions or no solutions at all, depending on the equations in the system and their relationships to each other. It is also possible for a non-homogeneous system to have a unique solution.

## 5. What are some applications of non-homogeneous systems?

Non-homogeneous systems are commonly used in various fields of science and engineering, such as physics, chemistry, and economics. They can be used to model real-world situations and solve problems related to systems of linear equations, such as finding unknown quantities or predicting outcomes.

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