### System of Linear Equations ConsistencyTo determine the conditions for the following system of linear equations to be consistent, the equations are given as:1. $-1x_1 + 4x_2 - 9x_3 = 9$2. $-3x_1 + 7x_2 + kx_3 = -2$3. $-6x_1 - 5x_2 + 6x_3 = 2$For the system to be consistent, it is stated that:\[ k \neq -\frac{4}{9} \]This means that the system will be inconsistent if $k$ is equal to $-\frac{4}{9}$.### InstructionsYou can enter your answer as a fraction to determine the value of $k$ that maintains the system's consistency.### InterfaceA box is provided for inputting the value of $k$. There is a "Check Answer" button below the input field, which allows you to verify your solution.

To determine the value of , for which the following system is consistent.

Answered: For the following system to be…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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### System of Linear Equations Consistency

To determine the conditions for the following system of linear equations to be consistent, the equations are given as:

1. $-1x_1 + 4x_2 - 9x_3 = 9$
2. $-3x_1 + 7x_2 + kx_3 = -2$
3. $-6x_1 - 5x_2 + 6x_3 = 2$

For the system to be consistent, it is stated that:

\[ k \neq -\frac{4}{9} \]

This means that the system will be inconsistent if $k$ is equal to $-\frac{4}{9}$.

### Instructions

You can enter your answer as a fraction to determine the value of $k$ that maintains the system's consistency.

### Interface

A box is provided for inputting the value of $k$. There is a "Check Answer" button below the input field, which allows you to verify your solution.

Expert Solution

Step 1: Define the problem.

To determine the value of $straight k$ , for which the following system is consistent.

$table row cell negative 1 straight x subscript 1 plus 4 straight x subscript 2 minus 9 straight x subscript 3 end cell equals 9 row cell negative 3 straight x subscript 1 plus 7 straight x subscript 2 plus kx subscript 3 end cell equals cell negative 2 end cell row cell negative 6 straight x subscript 1 minus 5 straight x subscript 2 plus 6 straight x subscript 3 end cell equals 2 end table$