Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations begin{cases}9x-3y+z=13 12x-8z=5 3x+4y-z =6 end{cases}

Determine whether the given set S is a subspace of the vector space V.
A. V=$P_5$, and S is the subset of $P_5$ consisting of those polynomials satisfying p(1)>p(0).
B. $V=R_3$, and S is the set of vectors $(x_1,x_2,x_3)$ in V satisfying $x_1-6x_2+x_3=5$.
C. $V=R^n$, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=$C^2(I)$, and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=$P_n$, and S is the subset of $P_n$ consisting of those polynomials satisfying p(0)=0.
G. $V=M_n(R)$, and S is the subset of all symmetric matrices

Determine whether the given $(2\times 3)(2\times 3)$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$x=at+b,y=ct+d,z=et+f$
$2x_1+x_2+x_3=3$
$-2x_1+x_2-x_3=1$

Given the linear system of equations:
$\{\begin{array}{l}{x}_1+x_2+x_3=n\\ x_1+x_2+x_3+x_4+x_5=3n\\ 2x_1+2x_2+2x_3+x_4+x_5+3x_6+3x_7+3x_8=10n\end{array}$
how many solutions are in $\mathbb{N}\cup \{0\}$?
The solution must not be using sum notation like $\sum y$.
I know how to find the number of solutions to the regular equations like $x_1+x_2+x_3+\cdots =n$ but I'm not sure how to do this for a system of equations. I thought of substituting some $x$'s:
$$\begin{gathered} x_1+x_2+x_3=3n-(x_4+x_5) \\ {x}_4+x_5=10n-2(x_1+x_2+x_3)-3(x_6+x_7+x_8) \\ ⟹x_1+x_2+x_3=3n-(10n-2(x_1+x_2+x_3)-3(x_6+x_7+x_8)) \\ ⟹x_1+x_2+x_3+3(x_6+x_7+x_8)=7n\ast \end{gathered}$$
As far as I understand finding the number of solutions for the system is equivalent to finding the number of solutions to the equation *.
The only next step from here I can think of is using generating functions:
$(1+x+x^2+\dots {)}^3(1+x^3+x^6+x^9+\dots {)}^3$
and we need to find the coefficient of $x^{7n}$.
From the closed form identities we have:
$\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{3-1+k}{k})x^k\cdot \sum _{i=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{3-1+i}{i})x^{3i}$
But I have no idea now how to find the coefficient of 7n from here and certainly not without using some kind of sum notation.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.

$\left[\begin{array}{ccccc}1& 0& -2& |& 6\\ 0& 1& 3& |& 1\end{array}\right]$

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. [101 011 000]

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
${\displaystyle \left[\begin{array}{cccccc}1& -1& 0& -2& 0& 0\\ 0& 0& 1& 2& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]}$

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y. or x, y, z. or $x_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{ccccc}1& 0& 0& 0& 1\\ 0& 1& 0& 0& 2\\ 0& 0& 1& 2& 3\end{array}\right]$