Chapter 11.2, Problem 30AYU

In Problems 27-38, 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.

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

To determine

To find: The system of equations corresponding to the given reduced row echelon augmented matrix and find the solution, if possible:

$\left(\begin{array}{ccc}1& 0& 4\\ 0& 1& 3\\ 0& 0& 0\end{array}|\begin{array}{c}4\\ 2\\ 0\end{array}\right)$

Answer to Problem 30AYU

Solution:

$x+4z=4$

$y+3z=2$

Consistent system of equations, infinitely many solutions.

Explanation of Solution

Given:

$\left(\begin{array}{ccc}1& 0& 4\\ 0& 1& 3\\ 0& 0& 0\end{array}|\begin{array}{c}4\\ 2\\ 0\end{array}\right)$

Calculation:

The system of equations corresponding to the given reduced row echelon augmented matrix is:

$x+4z=4$

$y+3z=2$

We find that the rank of the coefficient matrix is 2.

The rank of augmented matrix is $2<3$.

Both the ranks are equal and is equal to 2 (less than 3).

Hence the system of equations is consistent or has infinitely many solutions.