Chapter 10.5, Problem 38E

To determine

To calculate: Solution of the equations given using calculator.

Answer to Problem 38E

Solution of the equations is $x=8,y=4,z=2,w=1$

Explanation of Solution

Given information:

  $\{\begin{array}{l}x+y+z+w=15\\ x-y+z-w=5\\ x+2y+3z+4w=26\\ x-2y+3z-4w=2\end{array}$

Calculation:

Given equations are,

  $\{\begin{array}{l}x+y+z+w=15\\ x-y+z-w=5\\ x+2y+3z+4w=26\\ x-2y+3z-4w=2\end{array}$

Rewrite the system as a matrix equation of form $AX=B$

  $\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 1& -1\\ 1& 2& 3& 4\\ 1& -2& 3& -4\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ w\end{array}\right]=\left[\begin{array}{c}15\\ 5\\ 26\\ 2\end{array}\right]$

Where,

  $A=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 1& -1\\ 1& 2& 3& 4\\ 1& -2& 3& -4\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\\ w\end{array}\right],B=\left[\begin{array}{c}15\\ 5\\ 26\\ 2\end{array}\right]$

Need to solve for $X$ so, multiply both sides of equation by $A^{-1}$ , the inverse of quotient matrix.

Enter matrices in graphing calculator as below,

Press $2^{nd}+matrix$ and from the edit submenu select a matrix.

Enter dimensions of the matrix (rows and columns) then press ENTER.

Enter a value into each cell. Press ENTER to register each entry and to move the cursor to the next position.

Enter the matrix $B$ .

After each matrix, press $2^{nd}+$ Quit to return to home screen.

To find out the matrix $X$ ,

Find $A^{-1}B$

Go to $2^{nd}+$ matrix $+$ name and enter the matrix $A$ then press $X^{-1}$ and press $\times$ go $2^{nd}+$ matrix $+$ name and after that matrix $B$ and press enter button.

And value will get as:

  $x=8,y=4,z=2,w=1$ .