Problem 44 - College Algebra with Modeling and Visualization

Question

Represent the system of linear equations in the form \(A X=B\) \(4 x-3 y+2 z=8\) \(-x+4 y+3 z=2\) \(-2 x \quad-5 z=2\)

Step-by-Step Solution

Answer

The system is represented by \( A = \begin{bmatrix} 4 & -3 & 2 \\ -1 & 4 & 3 \\ -2 & 0 & -5 \end{bmatrix} \), \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \), and \( B = \begin{bmatrix} 8 \\ 2 \\ 2 \end{bmatrix} \).

Step 1: Identify Coefficients

First, identify the coefficients for each variable (x, y, z) in each of the equations. The first equation has coefficients 4, -3, 2. The second equation has coefficients -1, 4, 3. The third equation has coefficients -2, 0, -5 (note the absence of the y term, which means its coefficient is 0).

Step 2: Construct Matrix A

Place the coefficients of the variables into a matrix A: \[ A = \begin{bmatrix} 4 & -3 & 2 \ -1 & 4 & 3 \ -2 & 0 & -5 \end{bmatrix} \]

Step 3: Define Vector X

Vector X is simply the column vector of the variables for which we're solving: \[ X = \begin{bmatrix} x \ y \ z \end{bmatrix} \]

Step 4: Construct Matrix B

Matrix B contains the constants from the right side of each equation: \[ B = \begin{bmatrix} 8 \ 2 \ 2 \end{bmatrix} \]

Step 5: Form the Matrix Equation

Assemble the matrix equation \( AX = B \) by combining Matrix A, Vector X, and Matrix B: \[ \begin{bmatrix} 4 & -3 & 2 \ -1 & 4 & 3 \ -2 & 0 & -5 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 8 \ 2 \ 2 \end{bmatrix} \]

Key Concepts

Matrix Representation

Matrix representation is a method used in mathematics to organize and solve systems of linear equations. By placing coefficients and variables in a grid-like array, known as a matrix, we can visually handle complex equations and efficiently operate on them using matrix algebra. In the given exercise, the matrix representation of the system of equations allows us to handle all three equations simultaneously, rather than individually, leading to a more systematic approach to finding solutions.

Coefficients

Coefficients are the numerical values found before variables in algebraic expressions. They are crucial in forming a system of linear equations, as they determine how each variable contributes to the equation's outcome.
For example, in the equation \(4x - 3y + 2z = 8\), the coefficients for \(x\), \(y\), and \(z\) are 4, -3, and 2, respectively. These coefficients are assembled into a matrix, representing the relationships between variables.

• In the first equation, the coefficients are 4, -3, and 2.
• In the second equation, they are -1, 4, and 3.
• In the third equation, they are -2, 0 (as the \(y\) term is absent), and -5.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. It is a fundamental tool in solving systems of linear equations using matrices.
By understanding linear algebra, you can easily manipulate and solve equations using matrices. Linear algebra operations, such as matrix multiplication and inversion, allow for the efficient handling of multiple equations at once.
This approach provides powerful techniques for finding solutions, such as using determinant and inverse methods, which become feasible when equations are represented in matrix form.

Matrix Equation

A matrix equation is a way of writing a system of linear equations in a compact form, using matrices to represent the equations themselves.
In the matrix equation \(AX = B\),

• \(A\) is the matrix of coefficients.
• \(X\) is the vector of variables.
• \(B\) is the matrix (or vector) of constants.

This representation is incredibly useful as it simplifies complex algebraic manipulations and streamlines the computation process. Matrix equations are solved using different techniques in linear algebra, and they are foundational in computational methods.