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Chapter 9: Problem 15

Write the augmented matrix for each system of equations. \(x-2 y=4\) \(3 x+2 y=-5\)

Short Answer

Expert verified
The augmented matrix is \(\begin{bmatrix} 1 & -2 & | & 4 \ 3 & 2 & | & -5 \end{bmatrix}\).

Step by step solution

01

Identify Coefficients

First, write down the coefficients of each variable and the constants from both equations. For the first equation, the coefficients are 1, -2, and the constant is 4. For the second equation, the coefficients are 3, 2, and the constant is -5.
02

Write the Augmented Matrix

Combine these coefficients and constants into a single matrix. The augmented matrix places the coefficients in the first columns and the constants in the last column. The augmented matrix for this system is:ewline\(\begin{bmatrix} 1 & -2 & | & 4 \ 3 & 2 & | & -5 \end{bmatrix}\)

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Key Concepts

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

systems of equations
A system of equations consists of two or more equations that share the same set of variables. In our example, the system is comprised of two equations:
  • \[\begin{equation} x - 2y = 4 \end{equation}\]
  • \[\begin{equation} 3x + 2y = -5 \end{equation}\]

In these equations, we need to find values for x and y that satisfy both equations simultaneously. By solving a system of equations, we can pinpoint the exact values of these variables. This process is foundational in algebra and helps us understand relationships between variables in various fields like physics, economics, and engineering.
coefficients
Coefficients are the numerical factors that multiply the variables in an equation. In the given system, the coefficients are easy to spot. For the first equation, \(1x - 2y = 4 \), the coefficients are 1 and -2. Similarly, in the second equation, \(3x + 2y = -5 \), the coefficients are 3 and 2.
  • For \(1x - 2y = 4\), the numbers 1 and -2 are the coefficients of x and y, respectively.
  • For \(3x + 2y = -5 \), the numbers 3 and 2 are the coefficients of x and y, respectively.

These coefficients are crucial because they tell us how much of each variable is involved in the relationships described by each equation. When forming an augmented matrix, these coefficients are organized systematically.
matrix representation
Matrix representation is a way of displaying a system of equations. It uses a rectangular array of numbers to show all the coefficients and constants in a structured form. For our system of equations, we create an augmented matrix that includes both the coefficients and the constants.

Here's how we do it step by step:
  • Identify the coefficients: From the two equations, we have the coefficients 1, -2, for the first equation, and 3, 2 for the second. The constants are 4 and -5, respectively.
  • Form the matrix: The augmented matrix places the coefficients in the first columns and the constants in the last column. Our augmented matrix looks like this:
    \( \begin{bmatrix} 1 & -2 & | & 4 \ 3 & 2 & | & -5 \end{bmatrix} \)

By forming this matrix, we can use various techniques such as row operations for solving systems of equations, making it a powerful tool in linear algebra.

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Most popular questions from this chapter

Prove each of the following statements for any \(3 \times 3\) matrix \(A\). If all entries in any row or column of \(A\) are zero, then \(|A|=0\).

65\. \(1.5 x-5 y+3 z=16\) \(\begin{aligned} 2.25 x-4 y+z &=24 \\ 2 x+3.5 y-3 z &=-8 \end{aligned}\)

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Let \(A=\left[\begin{array}{ll}1 & 2 \\ 1 & 3\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{rr}3 & -2 \\ -1 & 1\end{array}\right] .\) Find \(A B\)

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