Boost Your Understanding of Linear Equation Forms

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.

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

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 $x_1,x_2,$from left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

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. 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⎤⎦⎥

Give an example of the row-echelon form of an augmented matrix that corresponds to an infinitely solvable system of linear equations.

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.

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

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.

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, x, y, and z cannot contain an equation in the form $y=mx+b$.

Substitute each point (-3, 5) and (2, -1) into the slope-intercept form of a linear equation to write a system of equations. Then use the system to find the equation of the line containing the two points.

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 x1,x2,$\dots$ from left to right.
$\left[\begin{array}{cccc}1& -3& 7& 1\\ 0& 1& 4& 0\\ 0& 0& 0& 1\end{array}\right]$