Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{r} x-5 y=5 \\ 4 x+3 y=6 \end{array}\right. $$

Short answer

[1 -5 | 5; 4 3 | 6]

Step-by-step solution

Write the System of Equations

The given system of equations is: \(x - 5y = 5\) and \(4x + 3y = 6\).

Express in Matrix Form

Rewrite the system in matrix form. The coefficients of the variables and the constants are arranged in matrix form as follows:\[\begin{bmatrix} 1 & -5 \ 4 & 3 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 \ 6 \end{bmatrix}\]

Form the Augmented Matrix

Create the augmented matrix by combining the coefficients matrix and the constants matrix: \[\begin{bmatrix} 1 & -5 & | & 5 \ 4 & 3 & | & 6 \end{bmatrix}\]

Key concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables.
The equations are often linear, but they can be nonlinear as well.
The main goal when working with a system of equations is to find the values of the variables that satisfy all equations simultaneously.

Consider the following system:
\(x - 5y = 5\) and \(4x + 3y = 6\).
Here, we have two linear equations with variables \(x\) and \(y\).

To solve this system, we need to find values for \(x\) and \(y\) such that both equations hold true at the same time.
For instance, if \(x = 1\) and \(y = 1\), do these values satisfy both equations? We can check by plugging these values into each equation:

• For the first equation: \(1 - 5(1) = -4\), which does not equal \(5\).
• For the second equation: \(4(1) + 3(1) = 7\), which does not equal \(6\).

Therefore, \(x = 1\) and \(y = 1\) is not a solution.
Finding the correct values requires methods such as substitution, elimination, or using an augmented matrix.

An augmented matrix is particularly useful for solving larger systems, but let's explore how it works with this example.

Matrix Form

The matrix form is a concise way to represent a system of linear equations.
In the matrix form, we separate the coefficients of variables and the constants.
For the given system:
\(x - 5y = 5\)
\(4x + 3y = 6\)
We create a coefficient matrix, a variable matrix, and a constants matrix:

• Coefficient Matrix: \(\begin{bmatrix} 1 & -5 \ 4 & 3 \end{bmatrix}\)
• Variable Matrix: \(\begin{bmatrix} x \ y \end{bmatrix}\)
• Constants Matrix: \(\begin{bmatrix} 5 \ 6 \end{bmatrix}\)

We then combine these matrices into a single matrix equation:
\[ \begin{bmatrix} 1 & -5 \ 4 & 3 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 \ 6 \end{bmatrix} \]
The matrix form simplifies the visualization and manipulation of systems of equations, especially when solving them using matrix operations.
Next, we will discuss how to form the augmented matrix from this representation.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces.
It plays a significant role in many areas of mathematics, science, and engineering.
When dealing with systems of equations, linear algebra provides powerful tools to solve them efficiently.

• Augmented Matrix: In linear algebra, an augmented matrix is a way to represent a system of linear equations. We append the constants to the coefficient matrix, separated by a vertical line:
  \[\begin{bmatrix} 1 & -5 & | & 5 \ 4 & 3 & | & 6 \end{bmatrix}\]
• Row Operations: These operations include swapping rows, multiplying a row by a non-zero constant, and adding or subtracting rows. These are used to simplify the matrix to row echelon form or reduced row echelon form, making it easier to solve the system.
• Solutions: The solution to a system can be found by transforming the augmented matrix into a form that clearly shows the values of the variables. If a unique solution exists, it can be read directly from the final matrix form.

Learning linear algebra and how to use matrix forms and operations can simplify complex problem-solving in mathematics and applied sciences.
In our equation system, the use of the augmented matrix streamlines the process of finding the values of \(x\) and \(y\).
By mastering these concepts, tackling systems of equations will become more intuitive and efficient.