Problem 43 - College Algebra with Modeling and Visualization

Question

Represent the system of linear equations in the form \(A X=B\) \(x-2 y+z=5\) \(3 y-z=6\) \(5 x-4 y-7 z=0\)

Step-by-Step Solution

Answer

The system in matrix form is \(AX = B\) with \(A = \begin{bmatrix} 1 & -2 & 1 \\ 0 & 3 & -1 \\ 5 & -4 & -7 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 5 \\ 6 \\ 0 \end{bmatrix}\).

Step 1: Identify the Variables

Identify the variables from the given system of equations. In this case, the variables are \(x\), \(y\), and \(z\).

Step 2: Write the Equations in Standard Form

The given system of equations can already be considered in standard form: 1. \(x - 2y + z = 5\)2. \(0x + 3y - z = 6\)3. \(5x - 4y - 7z = 0\)

Step 3: Extract the Coefficients

Identify the coefficients of each variable in each equation:- From equation 1: \(1, -2, 1\)- From equation 2: \(0, 3, -1\)- From equation 3: \(5, -4, -7\)

Step 4: Construct Matrix A

Form matrix \(A\) using the coefficients from each equation:\[A = \begin{bmatrix} 1 & -2 & 1 \ 0 & 3 & -1 \ 5 & -4 & -7 \end{bmatrix}\]

Step 5: Construct Matrix X

Construct matrix \(X\) with the variables \(x\), \(y\), and \(z\):\[X = \begin{bmatrix} x \ y \ z \end{bmatrix}\]

Step 6: Construct Matrix B

Form matrix \(B\) using the constants from the right-hand side of each equation:\[B = \begin{bmatrix} 5 \ 6 \ 0 \end{bmatrix}\]

Step 7: Write the System in Matrix Form

Combine the matrices into the form \(AX = B\). Thus, the system of equations can be represented as:\[\begin{bmatrix} 1 & -2 & 1 \ 0 & 3 & -1 \ 5 & -4 & -7 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 5 \ 6 \ 0 \end{bmatrix}\]

Key Concepts

Matrix Representation

In systems of linear equations, matrix representation is a powerful way to organize and solve equations. It transforms complex systems into a format that can be easily manipulated using matrix operations. In our exercise, we aim to express the given set of equations in the form \( A X = B \). This notation involves using matrices to represent the equations more compactly.
Here, matrix \( A \) contains all the coefficients from the left-hand side of the equations, matrix \( X \) represents the variables, and matrix \( B \) consists of the constants on the right-hand side:

• Matrix \( A \): Contains the numerical coefficients.
• Matrix \( X \): Holds the variables \( x, y, \) and \( z \).
• Matrix \( B \): Includes the numbers from the other side of the equations.

This method not only simplifies the visualization of linear equations but also enhances computational efficiency.

Coefficients

Coefficients are the numerical factors that multiply the variables in algebraic expressions. They are an essential component of linear equations, providing a way to scale the variables. In the system of equations we're examining:

• Equation 1: The coefficients are \( 1 \), \(-2 \), and \( 1 \) for \( x \), \( y \), and \( z \) respectively.
• Equation 2: The coefficients are \( 0 \), \( 3 \), and \(-1 \).
• Equation 3: The coefficients are \( 5 \), \(-4 \), and \(-7 \).

Recognizing coefficients enables us to construct Matrix \( A \), where each row is derived from the coefficients of the corresponding equation. This step is crucial for representing equations in matrix format and forms the basis for subsequent calculations.

Variables

Variables in a system of equations are the symbols used to represent unknown values. In our particular set of linear equations, the variables are \( x \), \( y \), and \( z \).
These variables are what we aim to solve for, using the system of equations.
The arrangement of these variables in matrix representation is systematic:

• Placed in matrix \( X \), representing the unknowns.
• Displayed in the same order across rows for consistency.

By isolating the variables into a vector (matrix \( X \)), we can perform operations on the matrix equation to find their values efficiently. It streamlines the solution process, aligning with methods like determinant calculations or using inverse matrices if applicable.

Standard Form

The standard form of a system of linear equations is crucial for matrix representation. It ensures that every equation lines up correctly, with variables on one side and constants on the other, and each term structured uniformly.
The typical format is:

• The left side contains variables, each associated with a coefficient.
• The right side consists of a constant term.

In our example, the equations are already in standard form, which simplifies the transition into matrix form. Each equation appears as \( ax + by + cz = d \), clearly showing coefficients leading the terms:
1. \( x - 2y + z = 5 \)
2. \( 0x + 3y - z = 6 \)
3. \( 5x - 4y - 7z = 0 \)
This consistency aids in extracting information systematically, ensuring accurate construction of the corresponding matrices \( A, X, \) and \( B \).