Q. 74 - Precalculus Enhanced with Graphing Utilities

Question

Use the idea behind Example  with a graphing utility to solve the following systems of equations. Round answers to two decimal places.

$\left\{\begin{array}{l}25x+61y-12z=25\\ 18x-12y+7z=10\\ 3x+4y-z=-4\end{array}\right.$

Step-by-Step Solution

Answer

Thus the values of variables are $x=0.54, y=0.01 \& z=0.052$

Step 1.Given information

Given equations $\begin{array}{l}25x+61y-12z=25\\ 18x-12y+7z=10\\ 3x+4y-z=-4\end{array}$

Step 2.Convert the equation

Convert these equation into matrix equation of the form $AX=B$

$\left[\begin{array}{ccc}25& 61& -12\\ 18& -2& 4\\ 8& 35& 21\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}25\\ 10\\ -4\end{array}\right]$

Step 3.Inverse matrix

To find the solution we find the inverse of the matrix  $\left[\begin{array}{ccc}25& 61& -12\\ 18& -2& 4\\ 8& 35& 21\end{array}\right]$

The inverse matrix =$\left[\begin{array}{ccc}0.00& 0.05& -0.01\\ 0.01& -0.02& 0.01\\ -0.02& 0.01& 0.03\end{array}\right]$

Step 4.Substitution

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{ccc}0.00& 0.05& -0.01\\ 0.01& -0.02& 0.01\\ -0.02& 0.01& 0.03\end{array}\right]\left[\begin{array}{c}25\\ 10\\ 4\end{array}\right]$

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0.54\\ 0.01\\ -0.52\end{array}\right]$

Step 5. The solution

Thus the values of variables are $x=0.54, y=0.01 \& z=0.052$