Q. 71 - Precalculus Enhanced with Graphing Utilities

Question

Use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.

$\left\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}\right.$

Step-by-Step Solution

Answer

On solving the system of equations, we get,

$$\begin{gathered} x=4.57, \\ y=-6.44, \\ z=-24.07 \end{gathered}$$

Step 1. Given information.

Consider the given question,

$\left\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}\right.$

Step 2. Assume the equations.

Consider the given question and assume,

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

The original system of equations can be written compactly as the matrix equation. Then,

$AX=B ...... \left(i\right)$

Step 3. Find the inverse matrix.

We need to find the inverse matrix of A,

$A^{-1}=\left[\begin{array}{ccc}-0.02& 0.02& 0.41\\ 0.06& 0.02& -0.57\\ 0.16& 0.12& -2.05\end{array}\right]$

Multiply each side of equation (i) by $A^{-1}$,

$$\begin{gathered} A^{-1}\left(AX\right)=A^{-1}B \\ \left(A^{-1}A\right)X=A^{-1}B \\ {I}_{3}X=A^{-1}B \\ X=A^{-1}B ...... \left(ii\right) \end{gathered}$$

Use equation (ii) to find X. This can be done using the graphing utility.

On enter the matrices A and B into a graphing utility, we get,

$$\begin{gathered} x=4.57, \\ y=-6.44, \\ z=-24.07 \end{gathered}$$