Chapter 8.1, Problem 25E

To determine

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

Answer to Problem 25E

The system of equation corresponding to the given matrix is

  $\begin{array}{c}9x+12y+3z=0\\ -2x+18y+5z+2w=10\\ x+7y-8z=-4\\ 3x+2z=-10\end{array}$ .

Explanation of Solution

Given information:

The given linear equations are

  $\left[\begin{array}{cccccc}9& 12& 3& 0& ⋮& 0\\ -2& 18& 5& 2& ⋮& 10\\ 1& 7& -8& 0& ⋮& -4\\ 3& 0& 2& 0& ⋮& -10\end{array}\right]$

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 $\left[\begin{array}{cccccc}9& 12& 3& 0& ⋮& 0\\ -2& 18& 5& 2& ⋮& 10\\ 1& 7& -8& 0& ⋮& -4\\ 3& 0& 2& 0& ⋮& -10\end{array}\right]$ .

We know that, in the augmented matrix, the numbers on the left side of the solid line form the coefficient matrix, with each column containing the coefficients of a variable of the system of equations and the column on the right side of the line contains the constant terms of the equations.

Hence, the system of linear equations represented by the above augmented matrix is

  $\begin{array}{c}9x+12y+3z+0w=0\\ -2x+18y+5z+2w=10\\ x+7y-8z+0w=-4\\ 3x+0y+2z+0w=-10\end{array}$

By simplifying further

  $\begin{array}{c}9x+12y+3z=0\\ -2x+18y+5z+2w=10\\ x+7y-8z=-4\\ 3x+2z=-10\end{array}$