Problem 39 - College Algebra with Modeling and Visualization

Question

Represent the system of linear equations in the form \(A X=B\) \(2 x-3 y=7\) \(-3 x-4 y=9\)

Step-by-Step Solution

Answer

System is represented as \( \begin{bmatrix} 2 & -3 \\ -3 & -4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 9 \end{bmatrix} \).

Step 1: Introduce the System of Equations

We are given a system of two linear equations: \[ 2x - 3y = 7 \] \[ -3x - 4y = 9 \] We need to represent this system in matrix form \( AX = B \).

Step 2: Formulate Matrix A

Matrix \( A \) is composed of the coefficients of \( x \) and \( y \) from the equations. From the first equation, the coefficients are 2 and -3, and from the second equation, the coefficients are -3 and -4. Thus, \[ A = \begin{bmatrix} 2 & -3 \ -3 & -4 \end{bmatrix} \]

Step 3: Formulate Matrix X

Matrix \( X \) represents the variables of the equations. For our system, it is the column matrix of the variables \( x \) and \( y \):\[ X = \begin{bmatrix} x \ y \end{bmatrix} \]

Step 4: Formulate Matrix B

Matrix \( B \) is made up of the constants from the right side of the equations. Here, these constants are 7 and 9. Therefore, \[ B = \begin{bmatrix} 7 \ 9 \end{bmatrix} \]

Step 5: Combine into Matrix Equation

Now, combine the matrices to form the equation \( AX = B \): \[ \begin{bmatrix} 2 & -3 \ -3 & -4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 7 \ 9 \end{bmatrix} \]This is the required matrix representation of the system of equations.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and often appear in the form \( ax + by = c \). In our exercise, two linear equations were provided:

• \( 2x - 3y = 7 \)
• \( -3x - 4y = 9 \)

Each equation represents a straight line on a Cartesian plane. Where the lines cross each other is the solution to the system if one exists. The coefficients (the numbers in front of \( x \) and \( y \)) and the constant terms (the numbers on the other side of the equation) are essential in formulating the corresponding matrices. Linear equations are characterized by variables raised to the first power, and their graphs are always straight lines. Understanding the concept of linear equations is crucial for solving systems of equations using matrix algebra.

Matrix Algebra

Matrix algebra provides a way to represent systems of equations in a compact and structured form. In our task, we converted the given system of linear equations into the matrix equation \( AX = B \):

• Matrix \( A \) represents the coefficients of the variables \( x \) and \( y \).
• Matrix \( X \) is a column matrix of the variables \( x \) and \( y \).
• Matrix \( B \) contains the constant terms from the equations.

Using matrices makes it easier to solve and manipulate systems of linear equations, especially when involving more variables or multiple equations. The operation rules for matrices might differ from regular arithmetic: they involve special processes like matrix multiplication and finding the inverse of a matrix.
Matrix algebra is powerful as it helps in finding solutions to systems of equations, determining when no solution exists, or when infinite solutions are possible.

Systems of Equations

A system of equations involves multiple equations that share some variables. In mathematics, there are often three possible outcomes when solving a system:

• A single solution where the lines intersect at one point.
• No solution where the lines are parallel and never intersect.
• Infinitely many solutions where the lines overlap completely.

In our example, the system consists of two equations:

• \( 2x - 3y = 7 \)
• \( -3x - 4y = 9 \)

Representing such systems with matrices like \( AX = B \) allows for easier analysis and solution computation, particularly as the number of equations and variables increases. Solving systems of equations with matrix algebra often involves techniques such as Gaussian elimination or using the inverse of a matrix, where applicable. This approach is especially valuable in various fields including engineering, physics, and computer science, where complex linear systems frequently arise.