Problem 30 - College Algebra

Question

In Exercises \(29-32,\) write each linear system as a matrix equation in the form \(A X=B\), where \(A\) is the coefficient matrix and \(B\) is the constant matrix. $$\left\\{\begin{array}{l}7 x+5 y=23 \\\3 x+2 y=10\end{array}\right.$$

Step-by-Step Solution

Answer

The given system of linear equations can be represented in matrix form as \( \[ \left[ \begin{array}{cc} 7 & 5 \\ 3 & 2 \end{array} \right] \[ \left[ \begin{array}{c} x \\ y \end{array} \right] \] = \[ \left[ \begin{array}{c} 23 \\ 10 \end{array} \right] \] \)

Step 1: Identify Matrices A, X, and B

In this exercise, matrix \(A\) is made up of the coefficients of the variables, \(X\) is the column matrix of the variables \(x\) and \(y\), and \(B\) is the column matrix of the constant terms. Therefore, matrix \(A\) is \( \[ \left[ \begin{array}{cc} 7 & 5 \\ 3 & 2 \end{array} \right] \] \), matrix \(X\) is \( \[ \left[ \begin{array}{c} x \\ y \end{array} \right] \] \), and matrix \(B\) is \( \[ \left[ \begin{array}{c} 23 \\ 10 \end{array} \right] \] \)

Step 2: Write System in Matrix Form

Now, put these matrices into the form \(AX = B\). The resulting matrix equation is \( \[ \left[ \begin{array}{cc} 7 & 5 \\ 3 & 2 \end{array} \right] \[ \left[ \begin{array}{c} x \\ y \end{array} \right] \] = \[ \left[ \begin{array}{c} 23 \\ 10 \end{array} \right] \] \)

Key Concepts

Linear Systems

A linear system is a collection of linear equations that involve the same set of variables. In our example, we have two equations:

• 7x + 5y = 23
• 3x + 2y = 10

These equations share the same variables, x and y, and we need to find the values that satisfy both equations simultaneously.
Linear systems often appear in algebra and can arise in various practical scenarios, such as when calculating intersections or optimizing solutions in economics and science.
To solve these systems, methods like graphing, substitution, elimination, or matrix operations are typically used. Matrix representation is a powerful method, especially for larger systems, as it allows for efficient computation and can be easily handled by computer algorithms.

Coefficient Matrix

The coefficient matrix, denoted as matrix A, is composed of all the coefficients of the variables in the linear system. In our example:

• The matrix for the coefficients is \(\left[\begin{array}{cc}7 & 5 \3 & 2\end{array}\right]\).

Each row corresponds to an equation in the system, and each column corresponds to a variable.
In general, when dealing with more variables or equations, this matrix helps to organize and simplify the calculations needed for finding the solution.
It is important to keep this matrix in the correct order, as switching rows or columns will reflect incorrect relationships between variables and their coefficients.

Constant Matrix

The constant matrix, known as matrix B, is composed of the constants from each equation in the system. For our exercise, matrix B is:

• \(\left[\begin{array}{c}23 \10\end{array}\right]\)

Each element in this matrix corresponds to the result of the respective equation when the variable terms have been isolated to one side.
This matrix is critical in expressing the linear system within the matrix equation format \(AX = B\).
Keeping the constants organized helps visualize and solve the system efficiently, especially when applying matrix operations to find solutions.

Matrix Representation

Matrix representation is a method of expressing linear systems as a single matrix equation, typically given as \(AX = B\). It enables operations that simplify the process of finding solutions.

• Matrix A represents the coefficients of the linear system.
• Matrix X is the variables column matrix, such as \(\left[\begin{array}{c}x \y\end{array}\right]\).
• Matrix B is the constants column matrix.

In matrix representation, each linear system can be succinctly expressed in a single equation rather than a set of equations.
This representation unlocks powerful tools such as matrix algebra, which can be utilized for more complex systems requiring computational approaches like matrix inversion or determinants to find solutions.
Using matrix representation not only aids in solution-finding but also provides a clearer understanding of the relationships and dynamics within the system of equations.