Chapter 11.4, Problem 71AYU

To determine

The solution of the system of equations $\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}$ by using graphing utility for the inverse of coefficient matrix.

Answer to Problem 71AYU

Solution:

The solution of the system of equations $\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}$ by using graphing utility for the inverse of coefficient matrix is $x=4.54,y=-6.42$ and $z=-24.08$ .

Explanation of Solution

Given information:

The system of equations, $\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}$ .

First write the given system in compact form $AX=B$ .

Here, $A=\left[\begin{array}{ccc}25& 61& -12\\ 18& -12& 7\\ 3& 4& -1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}10\\ -9\\ 12\end{array}\right]$ .

  $AX=B\Rightarrow X=A^{-1}B$ .

First find the inverse of the matrix $A$ by using calculator:

Step I: Press the ON key.

Step II: Press [2ND][ $x^{-1}$ ] to access the calculate menu.

Step III: Move curser to the “EDIT” menu at the top and press [ENTER].

Step IV: Select the size of the matrix that is number of rows by number of columns. Then plug the entries in that matrix form left to right. After each number press [ENTER] to go the next spot.

Step V: Press [2ND] [MODE] and go back into the matrix menu by clicking [2ND]

and [ $x^{-1}$ ] then press [1] and [ENTER].

Step VI: Press the inverse key [ $x^{-1}$ ] and press [ENTER].

The inverse is as below:

  $\left[\begin{array}{ccc}-0.02343& 0.01903& 0.41435\\ 0.05710& 0.01611& -0.57247\\ 0.15813& 0.12152& -2.04685\end{array}\right]$ .

By rounding numbers to two decimal places, the inverse is

  $\left[\begin{array}{ccc}-0.02& 0.02& 0.41\\ 0.06& 0.02& -0.57\\ 0.16& 0.12& -2.05\end{array}\right]$ .

To find the solution of system substitute $A^{-1}=\left[\begin{array}{ccc}-0.02& 0.02& 0.41\\ 0.06& 0.02& -0.57\\ 0.16& 0.12& -2.05\end{array}\right]$ and $B=\left[\begin{array}{c}10\\ -9\\ 12\end{array}\right]$ in $X=A^{-1}B$ .

  $\Rightarrow X=\left[\begin{array}{ccc}-0.02& 0.02& 0.41\\ 0.06& 0.02& -0.57\\ 0.16& 0.12& -2.05\end{array}\right]\left[\begin{array}{c}10\\ -9\\ 12\end{array}\right]$

Perform the matrix multiplication.

It gives the solution of system as

  $X=\left[\begin{array}{c}4.54\\ -6.42\\ -24.08\end{array}\right]$ .

Thus, the solution of the system of equations $\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}$ by using graphing utility for the inverse of coefficient matrix.is $x=4.54,y=-6.42$ and $z=-24.08$ .