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Chapter 9: Problem 14

Write the system of equations that corresponds to the augmented matrix. $$\left[\begin{array}{rrr|r} -1 & -2 & 3 & 6 \\ 0 & 4 & 1 & 2 \\ 2 & -1 & 0 & 9 \end{array}\right]$$

Short Answer

Expert verified
The system of equations is: \[-x - 2y + 3z = 6\]\[4y + z = 2\]\[2x - y = 9\]

Step by step solution

01

- Understand the Augmented Matrix Format

An augmented matrix represents a system of linear equations. Each row corresponds to one equation, and each column (excluding the last column after the vertical bar) corresponds to a variable coefficient.
02

- Identify Variables

Assign variables to the columns: let the first column be for variable x, the second for variable y, and the third for variable z.
03

- Write the First Equation

The first row of the augmented matrix corresponds to the coefficients of the first equation. Thus, \[-1x - 2y + 3z = 6\]This gives the first equation: \[-x - 2y + 3z = 6\]
04

- Write the Second Equation

The second row corresponds to the second equation. Thus, \[0x + 4y + 1z = 2\]This simplifies to the second equation: \[4y + z = 2\]
05

- Write the Third Equation

The third row corresponds to the third equation. Thus, \[2x - 1y + 0z = 9\]This simplifies to the third equation: \[2x - y = 9\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Augmented Matrix
An augmented matrix is a compact and organized way to represent a system of linear equations. The vertical bar in an augmented matrix acts as a visual divider, separating the coefficients of the variables from the constants on the right side of the equations. Each row in the matrix corresponds to a single linear equation, while each column prior to the vertical bar corresponds to the coefficients of those variables (e.g., x, y, and z). Using an augmented matrix can simplify solving systems of equations, especially when using methods like Gaussian elimination.
Variable Coefficients
Variable coefficients are the numerical values multiplying the variables in each equation of a system. In a matrix, these coefficients are aligned in columns. For example, in the augmented matrix shown in the exercise, the first column contains the coefficients for variable x, the second for variable y, and the third for variable z. Understanding how to correctly identify and position these coefficients is crucial to translating the augmented matrix back into its corresponding system of equations.
System of Equations
A system of equations is a collection of two or more linear equations involving the same set of variables. The goal is to find the values of these variables that satisfy all the equations simultaneously. For instance, the given augmented matrix in the exercise represents the system of equations:
  • -x - 2y + 3z = 6
  • 4y + z = 2
  • 2x - y = 9
By solving this system, we determine the particular values of x, y, and z that make each of these equations true at the same time. There are various methods to solve systems of equations, such as substitution, elimination, and matrix operations.

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Most popular questions from this chapter

Solve the system of equations using the inverse of the coefficient matrix of the equivalent matrix equation. $$\begin{array}{c}2 x+3 y+4 z=2 \\\x-4 y+3 z=2 \\\5 x+y+z=-4\end{array}$$

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