Chapter 11.2, Problem 72AYU

In problems 39-74, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

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

To determine

To find: The solution to the given system of equations using matrices.

Answer to Problem 72AYU

Solution:

Consistent system, infinitely many solutions.

Explanation of Solution

Given:

$-4x+y=5$

$2x-y+z-w=5$

$z+w=4$

Calculation:

To solve a system of three equations in $x,y$ and $z$ using matrices:

Step1: Write the corresponding matrix associated with the system of equations.

Step2: Use elementary row operations to get equivalent matrix of the form:

$\left[\begin{array}{ccc}1& a& b\\ 0& 1& c\\ 0& 0& 1\end{array}|\begin{array}{c}d\\ e\\ f\end{array}\right]$; where $a,b,c,d,e,f$ are constants.

Step3: Solve for $x,y$ and $z$.

$-4x+y=5$

$2x-y+z-w=5$

$z+w=4$

Here the number of variables is 4, whereas the number of equations is 3.

Since the number of equations is less than the number of variables, therefore the given system of equations is consistent and has infinitely many solutions.