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Chapter 9: Problem 12

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

Short Answer

Expert verified
The system is:\begin{cases} x + 2y = -6 \ 4x + y = -3 \end{cases}

Step by step solution

01

Identify the matrix layout

The given matrix \[ \begin{array}{ll|l} 1 & 2 & -6 \ 4 & 1 & -3 \end{array} \] represents the coefficients and constants of a system of linear equations. The columns before the vertical line represent the coefficients of the variables, and the column after the vertical line represents the constants.
02

Assign variables to columns

Assign variables to each column before the vertical line. Let's use x and y. The system can be written as: \[ \begin{array}{ll} 1x & + 2y & = -6 \ 4x & + 1y & = -3 \end{array} \]
03

Write the first equation

The first row corresponds to the equation with the coefficients from that row and the constant to the right of the vertical line: \[ 1x + 2y = -6 \]
04

Write the second equation

The second row corresponds to the equation with the coefficients from that row and the constant to the right of the vertical line: \[ 4x + 1y = -3 \]
05

Write the system

Combine the equations from the previous steps to form the complete system of linear equations: \[ \begin{cases} x + 2y = -6 \ 4x + y = -3 \end{cases} \]

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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 way to represent a system of linear equations. It combines the coefficients and constants from the equations into a single matrix. This allows for easier manipulation of the system using matrix operations. In an augmented matrix, the columns to the left of the vertical line contain the coefficients of the variables, and the column to the right of the vertical line contains the constants.
linear equations
Linear equations are equations where each term is either a constant or the product of a constant and a single variable. They form straight lines when graphed on a coordinate plane. A system of linear equations is a set of equations with the same variables. For example, the system derived from our augmented matrix is: \[ \begin{cases} x + 2y = -6 \ 4x + y = -3 \end{cases} \]. Solving this system can help find the values of the variables that satisfy all equations simultaneously.
variable assignment
Variable assignment is the process of designating a variable to each column of coefficients in the augmented matrix. This step is crucial for interpreting the matrix correctly. In our example, we assigned the variables 'x' and 'y' to the two columns from left to right. This lets us translate the rows of the matrix back into linear equations with these variables, forming \[ 1x + 2y = -6 \ 4x + 1y = -3 \]. This approach keeps the relationship between the variables and their coefficients clear.
coefficients
Coefficients are the numerical factors that multiply variables in linear equations. In the context of an augmented matrix, they appear as the elements in the columns before the vertical line. For our example, the coefficients are \[ \begin{array}{ll} 1 \text{ and } 2 \text{ in the first row and } 4 \text{ and } 1 \text{ in the second row.} \ \text{These coefficients help to link each variable to the appropriate constant term, forming the equations:} \begin{cases} x + 2y = -6 \ 4x + y = -3 \end{cases} \ \text{Understanding the role of coefficients is essential for solving or simplifying the system of equations.} \end{array} \]

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Most popular questions from this chapter

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