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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 71 (page 756)

Use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$\{\begin{cases} 25 x + 61 y - 12 z = 10 \\ 18 x - 12 y + 7 z = - 9 \\ 3 x + 4 y - z = 12 \end{cases}$

Short Answer

Expert verified

On solving the system of equations, we get,

$x = 4.57, y = - 6.44, z = - 24.07$

Step by step solution

01

Step 1. Given information.

Consider the given question,

$\{\begin{cases} 25 x + 61 y - 12 z = 10 \\ 18 x - 12 y + 7 z = - 9 \\ 3 x + 4 y - z = 12 \end{cases}$

02

Step 2. Assume the equations.

Consider the given question and assume,

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

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

$A X = B . . . . . . (i)$

03

Step 3. Find the inverse matrix.

We need to find the inverse matrix of A,

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

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

localid="1647582875252" $A^{- 1} (A X) = A^{- 1} B (A^{- 1} A) X = A^{- 1} B I_{3} X = A^{- 1} B X = A^{- 1} B . . . . . . (i i)$

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,

$x = 4.57, y = - 6.44, z = - 24.07$

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