Simplifying Linear Algebra: Techniques, Solutions, and Examples

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)
$x+2y-z=2,$
$2x-2y+z=1$

Vector Cross Product
Let vectors $A=(1,0,-3),B=(-2,5,1),andC=(3,1,1)$. Calculate the following, expressing your answers as ordered triples (three comma-separated numbers).
(C) $(2\overline{B})(3\overline{C})$
(D) $(\overline{B})(\overline{C})$
(E) $\overrightarrow{A}(\overrightarrow{B}\times \overrightarrow{C})$
(F)If ${\overline{v}}_1\text{ and }{\overline{v}}_2$ are perpendicular, $|{\overline{v}}_1\times {\overline{v}}_2|$
(G) If ${\overline{v}}_1\text{ and }{\overline{v}}_2$ are parallel, $|{\overline{v}}_1\times {\overline{v}}_2|$

Refer to the system of linear equations $\{\begin{array}{l}-2x+3y=5\\ 6x+7y=4\end{array}$ Is the augmented matrix row-equivalent to its reduced row-echelon form?

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution.

$\left[\begin{array}{ccc}1& -4& 5\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. $\left[\begin{array}{ccc}1& -2& 6\\ 0& 0& 0\end{array}\right]$

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

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2$... from left to right.

$\left[\begin{array}{ccccc}1& 0& 0& -7& 8\\ 0& 1& 0& 3& 2\\ 0& 0& 1& 1& -5\end{array}\right]$

The row echelon form of a system of linear equations is given.

(a)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.

(b)Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

$\left[\begin{array}{cccc}0& -3& |& -4\\ 0& 1& |& 0\end{array}\right]$

Each of the matrices is the final matrix form for a system of two linear equations in the variables $x_1$ and $x_2$. Write the solution of the system.

$\left[\begin{array}{cccc}1& 0& |& 3\\ 0& 1& |& -5\end{array}\right]$

Write the vector form of the general solution of the given system of linear equations.
$x_1+3x_2+2x_4=0$
$x_3-6x_4=0$

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;x,y; or x,y,z;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. ⎡⎣⎢100010430420⎤⎦⎥