A system of linear equations is

**homogeneous** if we can write the matrix equation in the form

$Ax=0$.

A system of linear equations is

**nonhomogeneous** if we can write the matrix equation in the form

$Ax=b$.

We can express solution sets of linear systems in

**parametric vector form**.

Here are the types of solutions a homogeneous system can have in parametric vector form:

1. With 1 free variable:

$x=tv$
2. With 2 free variables:

$x=su+tv$
3. With n free variables:

$x=av_1+bv_2+\cdots+nv_n$
Here are the types of solutions a nonhomogeneous system can have in a parametric vector form:

1. With 1 free variable:

$x=p+tv$
2. With 2 free variables:

$x=p+su+tv$
3. With n free variables:

$x=p+av_1+bv_2+\cdots+nv_n$