Boost Your Understanding of Linear Equation Forms

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. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{cccc}1& 2& -1& 0\\ 0& 1& -1& 1\\ 0& 0& 0& 2\end{array}\right]$

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $\left[\begin{array}{cccccc}1& 0& -1& -2& -3& 1\\ 0& 1& 2& 3& 4& 0\end{array}\right]$

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $\left[\begin{array}{cccccc}1& 0& -1& -2& -3& 0\\ 0& 1& 2& 3& 4& 0\end{array}\right]$

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

The given matrix is the augmented matrix for a system of linear equations $[\begin{array}{}1& 0& -1& 0& -1& -2& 0\\ 0& 1& 2& 0& 1& 2& 0\\ 0& 0& 0& 1& 1& 1& 0\end{array}]$

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 ${\displaystyle x_1,x_2,\dots }$ from left to right.
$\left[\begin{array}{ccccc}1& 1& -3& 2& 1\\ 0& 1& 4& 0& 3\\ 0& 0& 0& 1& 2\end{array}\right]$

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 $x_1,x_2,...x,...$ ,… from left to right. 

Let $AX=B$ be a system of linear equations, where A is an $m\times nm\times n$ matrix, X is an n-vector, and $BB$ is an m-vector. Assume that there is one solution $X=X0$. Show that every solution is of the form $X0+Y$, where Y is a solution of the homogeneous system $AY=0$, and conversely any vector of the form $X0+Y$ is a solution.

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

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

The row echelon form of a system of linear equations is given.
(a)Write the system of equations corresponding to the given matrix.Usex,y;orx,y,z;or $x_1,x_2,x_3,x_4$ asvariables.
(b)Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $\left[\begin{array}{ccc}1& 2& 5\\ 0& 1& -1\end{array}\right]$

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