Problem 42 - College Algebra with Modeling and Visualization

Question

Represent the system of linear equations in the form \(A X=B\) \(-1.1 x+3.2 y=-2.7\) \(5.6 x-3.8 y=-3.0\)

Step-by-Step Solution

Answer

The system is represented as \(AX = B\) where \(A = \begin{pmatrix}-1.1 & 3.2 \\ 5.6 & -3.8\end{pmatrix}\), \(X = \begin{pmatrix}x \\ y\end{pmatrix}\), and \(B = \begin{pmatrix}-2.7 \\ -3.0\end{pmatrix}\).

Step 1: Identify the Coefficients

First, we need to identify the coefficients of the variables from the given equations. The equations are: 1. \(-1.1x + 3.2y = -2.7\)2. \(5.6x - 3.8y = -3.0\).The coefficients for the first equation are -1.1 for \(x\) and 3.2 for \(y\). For the second equation, the coefficients are 5.6 for \(x\) and -3.8 for \(y\).

Step 2: Write the Coefficient Matrix

Using the coefficients identified in step 1, create the coefficient matrix \(A\). Thus, the matrix \(A\) is: \[A = \begin{pmatrix}-1.1 & 3.2 \5.6 & -3.8\end{pmatrix}\]

Step 3: Define the Variable Matrix

Next, define the variable matrix \(X\) using the variables from the equations. There are two variables, \(x\) and \(y\), so the matrix \(X\) will be: \[X = \begin{pmatrix} x \ y \end{pmatrix}\]

Step 4: Write the Constant Matrix

Identify the constant terms on the right side of the equations: \(-2.7\) and \(-3.0\). This forms the constant matrix \(B\):\[B = \begin{pmatrix} -2.7 \ -3.0 \end{pmatrix}\]

Step 5: Formulate the System as AX = B

Combine the matrices to express the system of equations in the desired form:\[\begin{pmatrix} -1.1 & 3.2 \ 5.6 & -3.8 \end{pmatrix}\begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -2.7 \ -3.0 \end{pmatrix}\]This represents the original system of linear equations in the standard matrix form \(AX = B\).

Key Concepts

Matrix Representation

When dealing with a system of linear equations, organizing them into a matrix form can simplify both understanding and solving the system. Think of matrices as neat rectangular arrays that help package information efficiently.
Imagine your equations as puzzles waiting to be solved. The matrix representation organizes this puzzle by gathering like terms into rows and columns, enabling easy manipulation and solution finding.
This method standardizes how we understand and solve simultaneous equations, making it a powerful tool.

• The equation structure follows the form of \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constant matrix.
• The purpose is to compact several linear equations into a simplified matrix equation that can be handled much like a single equation.

With this setup, you can deploy techniques such as matrix inversion or row operations to find solutions, making them accessible even at higher complexity levels.

Coefficient Matrix

The coefficient matrix is a crucial component in the matrix form of equations. It essentially houses all the coefficients of the variables from the system of equations.
In the given example, your coefficient matrix, denoted as \(A\), is \(\begin{pmatrix}-1.1 & 3.2 \ 5.6 & -3.8\end{pmatrix}\). Each element in this matrix corresponds to the numeric multiples of your variables from the equations.
By organizing the coefficients in this matrix form:

• Row 1 elements \(-1.1\) and \(3.2\) represent the coefficients of \(x\) and \(y\) from the first equation.
• Row 2 elements \(5.6\) and \(-3.8\) stem from the second equation for \(x\) and \(y\).

This organization allows you to quickly see and use these coefficients in solving methods such as Gaussian elimination or matrix inverses, making the approach efficient and universally applicable.

Variable Matrix

Variables are the unknowns you aim to solve in linear equations, and the variable matrix helps manage these elements neatly. In our case, this matrix is represented as \(X\) and is given by \(\begin{pmatrix} x \ y \end{pmatrix}\).
Here’s why using a variable matrix is effective:

• It isolates the unknowns \(x\) and \(y\) from their coefficients and constants.
• Keeping them together provides a clear structure that makes the math operations more straightforward.

Properly using this matrix allows methods like substitution and elimination to be applied on a broader range of equations efficiently, emphasizing the system's unknowns and how they interact with coefficients in matrix operations.

Constant Matrix

The constant matrix represents the solutions from individual equations when all the variables and their coefficients are summed up and calculated. This is denoted by \(B\), and for our exercise, is represented as \(\begin{pmatrix} -2.7 \ -3.0 \end{pmatrix}\).
This matrix takes all the constant terms on the right side of the equations and organizes them as:

• The constant \(-2.7\) from the first equation relating to its respective coefficients and variables.
• The constant \(-3.0\) from the second equation.

By compiling them into this neat column matrix, they stand ready to fulfill any algebraic need in solving \(AX = B\), maintaining the balance of the equations and ensuring that any solutions reached are accurate reflections of the original system formulated.