Chapter 8.1, Problem 26E

To determine

To write: the system of linear equations represented by the augmented matrix.

Answer to Problem 26E

The system of equation corresponding to the given matrix is

  $\begin{array}{c}6x+2y-z-5w=-25\\ -x+7z+3w=7\\ 4x-y-10z+6w=23\\ 8y+z-11w=-21\end{array}$ .

Explanation of Solution

Given information:

The given linear equations are

  $\left[\begin{array}{cccccc}6& 2& -1& -5& ⋮& 25\\ -1& 0& 7& 3& ⋮& 7\\ 4& -1& -10& 6& ⋮& 23\\ 0& 8& 1& -11& ⋮& -21\end{array}\right]$

Properties used:

An augmented matrix of the form

  $\left[\begin{array}{cccc}a& b& ⋮& u\\ c& d& ⋮& v\end{array}\right]$ has system of equation of the form

  $\{\begin{array}{l}ax+by=u\\ cx+dy=v\end{array}$

Calculation:

Consider the augmented matrix $\left[\begin{array}{cccccc}6& 2& -1& -5& ⋮& 25\\ -1& 0& 7& 3& ⋮& 7\\ 4& -1& -10& 6& ⋮& 23\\ 0& 8& 1& -11& ⋮& -21\end{array}\right]$ .

An augment matrix is the matrix derived from a system of linear equations by writing the coefficients and the constant terms of the equations. Constant terms are written on the right extreme.

Here since augmented matrix has three columns on left side of dots, use three variables.

Take three variables $x,y$ and $z$

The elements on left side of dots of augmented matrix are coefficient of variables and right side elements are constant terms.

Hence linear equations are,

  $\begin{array}{c}6x+2y-z-5w=-25\\ -x+7z+3w=7\\ 4x-y-10z+6w=23\\ 8y+z-11w=-21\end{array}$