Chapter 11.2, Problem 32AYU

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.

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

To determine

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

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

Answer to Problem 32AYU

Solution:

$x+y+2w=3$

$y+z+5w=2$

$x+z+3w=1$

Consistent system of equations, infinitely many solutions.

Explanation of Solution

Given:

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

Calculation:

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

$x+y+2w=3$

$y+z+5w=2$

$x+z+3w=1$

$(R_1=r_1+r_2;R_2=r_2+r_3;R_3=r_3+r_1)$

Here we see that the number of variables is 4 and the number of equations is 3.

The number of variables is more than the number of equations.

Thus the above system of equations has infinitely many solutions.

Hence the above system is consistent.