Simplifying Linear Algebra: Techniques, Solutions, and Examples

$5y+3=2y-3x+5$
$y=$ ___

Find $6^3\times 6^{-2}$

Solve the system of equations

$\{\begin{array}{l}2x+3y=5\\ 5x-4y=2\end{array}$

$(792)(5.19\times {10}^{-}5)$

Write the vector form of the general solution of the given system of linear equations.
$x_1+4x_4=0$
$x_2-2x_4$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given row echelon form. Solve the system by back substitution. Assume that the variables are named$x_1,x_2,\dots$ from left to right.
$\left[\begin{array}{cccc}1& -3& 7& 1\\ 0& 1& 4& 0\\ 0& 0& 0& 1\end{array}\right]$

Find a nonzero vector in Nul A and a nonzero vector in Col A for the matrix A below.
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$
In Nul A, locate a nonzero vector.
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Using T defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique.
$A\left[\begin{array}{ccc}1& -5& -7\\ -3& 7& 5\end{array}\right],B=\left[\begin{array}{c}-2\\ -2\end{array}\right]$

Solve the system by clennaton

$\{\begin{array}{l}2x-y=0\\ 3x-2y=-3\end{array}$
The solution is____

(a) Find a nonzero vector orthogonal l to the plane the points P, Q, and R, and
(b) find the area of triangle PQR $P(1,0,1),Q(-2,1,3),R(4,2,5)$

The set $B=\{1+t^2,t+t^2,1+2t+t^2\}$ is a basis for ${\displaystyle {\mathbb{P}}_2}$. Find the coordinate vector of ${\displaystyle p\left(t\right)=1+4t+7t^2}$ relative to B.

Determine if the columns of the matrix form a linearly independent set. Justify each answer.
$\left[\begin{array}{cccc}1& 4& -3& 0\\ -2& -7& 5& 1\\ -4& -5& 7& 5\end{array}\right]$

Reduce the system of linear equations to upper triangular form and solve.
$\{\begin{array}{l}3x-y=1\\ -3x+y=4\end{array}$

Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
$\left[\begin{array}{ccc}1& 3& -2\\ -4& h& 8\end{array}\right]$

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 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.

$\left[\begin{array}{cccc}1& -2& 0& -4\\ 0& 0& 1& 3\\ 0& 0& 0& 0\end{array}\right]$

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.

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