Problem 30 - Algebra and Trigonometry

Question

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 matrix equation \( AX = B \) is \[ \begin{bmatrix} 7 & 5 \ 3 & 2 \end{bmatrix} X = \begin{bmatrix} 23 \ 10 \end{bmatrix} \]

Step 1: Identify the Coefficients and Constants

The first equation is \( 7x + 5y = 23 \). Here, 7 and 5 are the coefficients of \( x \) and \( y \) respectively, and 23 is the constant. For the second equation \( 3x + 2y = 10 \), 3 and 2 are the coefficients of \( x \) and \( y \) respectively, and 10 is the constant.

Step 2: Construct the Coefficient Matrix A

The elements of our coefficient matrix \( A \) should be the coefficients of \( x \) and \( y \) from our system of equations. So, the coefficient matrix \( A \) is \[ \begin{bmatrix} 7 & 5 \ 3 & 2 \end{bmatrix} \]

Step 3: Construct the Constant Matrix B

The constant matrix \( B \) should contain the constants from our system of equations. So, the constant matrix \( B \) is \[ \begin{bmatrix} 23 \ 10 \end{bmatrix} \]

Step 4: Formulate the Matrix Equation AX = B

Having both matrix \( A \) and matrix \( B \), we can now formulate the matrix equation \( AX = B \). Thus, the matrix equation is \[ \begin{bmatrix} 7 & 5 \ 3 & 2 \end{bmatrix} X = \begin{bmatrix} 23 \ 10 \end{bmatrix} \]

Key Concepts

Matrix Equation

A matrix equation is a way of representing a system of linear equations in a compact form using matrices. Let's break down the concept, especially in the context of the linear system from our exercise.
By using matrices, we can succinctly represent the relationship between variables and constants. In the given exercise, you encounter the system of equations:

• 7x + 5y = 23
• 3x + 2y = 10

By arranging these into a matrix equation, we can write it as: \(A \cdot X = B\).
Here:

• \(A\) is a matrix containing the coefficients of the variables \(x\) and \(y\).
• \(X\) is a column matrix for the variables, typically denoted as \(\begin{bmatrix} x \ y \end{bmatrix}\).
• \(B\) is a matrix containing the constant terms from the equations.

This equation provides a structured way to handle multiple equations, simplifying the solution process using matrix operations.

Coefficient Matrix

In a linear system like the one in our exercise, the coefficient matrix is composed of the coefficients of the variables from the equations. Let's see how it works. For our stated system of equations:

• 7x + 5y = 23
• 3x + 2y = 10

The coefficients that accompany the variables \(x\) and \(y\) are essential for forming the coefficient matrix. Here's how it is structured:

• The first row corresponds to the coefficients from the first equation: 7 and 5.
• The second row draws from the second equation: 3 and 2.

Putting these together gives us the coefficient matrix \(A\):\[A = \begin{bmatrix} 7 & 5 \ 3 & 2 \end{bmatrix}\] This matrix serves as the foundation for analyzing the linear system. It's crucial for performing calculations like matrix multiplication, which is often used to find the variable values.

Constant Matrix

When handling linear systems, the constant matrix plays a crucial role alongside the coefficient matrix. The constant matrix comprises the constant terms from each of the equations in the system. For the given example:

• 7x + 5y = 23
• 3x + 2y = 10

The constants located on the right side of the equations are extracted to form the constant matrix. For this example, the matrix is structured as:\[B = \begin{bmatrix} 23 \ 10 \end{bmatrix}\]This vertical arrangement aligns with the corresponding equations' constants in the order they are written. The constant matrix helps in considering the equations' total values during the matrix multiplication process, facilitating the resolution of the variables like \(x\) and \(y\). Integrating the constant matrix with the coefficient matrix gives a complete view of the system, setting the stage for solution techniques like matrix equations and inverses.