Question

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

$\left\{\begin{array}{l}25x+61y-12z=21\\ 18x-12y+7z=7\\ 3x+4y-z=-2\end{array}\right.$

Short answer

On solving the system of equations, we get,

$$\begin{gathered} x=-1.1, \\ y=2.5, \\ z=8.3 \end{gathered}$$

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\left\{\begin{array}{l}25x+61y-12z=21\\ 18x-12y+7z=7\\ 3x+4y-z=-2\end{array}\right.$

Step 2. Assume the equations.

Consider the given question and assume,

$$\begin{gathered} A=\left[\begin{array}{ccc}25& 61& -12\\ 18& -12& 7\\ 3& 4& -1\end{array}\right], \\ X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right] \\ B=\left[\begin{array}{c}21\\ 7\\ -2\end{array}\right] \end{gathered}$$

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 $A^{-1}$,

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

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

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

$$\begin{gathered} x=-1.1, \\ y=2.5, \\ z=8.3 \end{gathered}$$