Problem 29 - College Algebra

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. $$ \begin{aligned} &6 x+5 y=13\\\ &5 x+4 y=10 \end{aligned} $$

Step-by-Step Solution

Answer

The matrix equation corresponding to the given system of equations is [6,5;5,4][x;y]=[13;10]

Step 1: Identify Matrix A

Matrix A is formed by the coefficients of the variables in the system of equations. For the given system, the coefficients are the numbers in front of variables x and y in both equations. So, matrix A would be [6,5;5,4]

Step 2: Identify Matrix B

Matrix B is formed by the constant terms in the system of equations. These are the numbers on the other side of the equal sign in both equations. In this case the constants are 13 and 10. So, Matrix B would be [13;10]

Step 3: Write the Matrix Equation

The system of equations in matrix form is AX = B, where X is a column matrix [x; y]. So, the given system of equations in matrix form would be [6,5;5,4][x;y]=[13;10]

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. A key operation in linear algebra is solving systems of linear equations, which involves finding the values of variables that make multiple algebraic equations true simultaneously.

Understanding linear algebra is essential for various fields, including engineering, physics, computer science, and more, as it provides a framework for handling many dimensions of data and the relationships between them. It allows for the simplification of complex problems by representing them in a concise, powerful way using matrices and vectors.

System of Linear Equations

A system of linear equations consists of two or more linear equations that are solved simultaneously. The goal is to find a common solution, that is, a set of values for the variables that will satisfy all the equations at once. In the given example, we have two equations with two variables each:

• 6x + 5y = 13
• 5x + 4y = 10

These can be thought of graphically as lines on a plane, and the common solution would be the point where these lines intersect. Solving such a system algebraically can be laborious by hand, but converting it to a matrix equation simplifies the process, especially when dealing with larger systems.

Coefficient Matrix

In the context of a system of linear equations, the coefficient matrix, often denoted as matrix A, is a matrix composed of the coefficients of the variables from the system. For the given system, we identify the coefficients of 'x' and 'y' from both equations and form the following matrix A:
\[ A = \begin{bmatrix}6 & 5 \5 & 4\end{bmatrix} \]
The coefficient matrix can be seen as a tool that captures all the variable coefficients of the system in a structured way. It’s the heart of the matrix equation, since it holds the values that, when multiplied by the variable matrix, lead to the results on the other side of the equations.

Constant Matrix

The constant matrix, typically signified as matrix B, incorporates the constants from the right-hand side of each equation in the system of linear equations. In our example, the constants are 13 and 10 from the respective equations:
\[ B = \begin{bmatrix}13 \10\end{bmatrix} \]
The importance of the constant matrix lies in its ability to represent the outcome or the results that we want to achieve once we plug in our solution for the variables. When we place matrix A and matrix B side by side in the matrix equation \( A \boldsymbol{X} = B \), this visually demonstrates the relationship between the coefficients of the variables and the outcomes they’re supposed to lead to upon finding the solution \( \boldsymbol{X} \).