Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{l} 3 x+4 y=7 \\ 4 x-2 y=5 \end{array}\right. $$

Short answer

The augmented matrix is \[ \begin{pmatrix} 3 & 4 & \big| & 7 \ 4 & -2 & \big| & 5 \ }\]

Step-by-step solution

- Write Down the Coefficients

First, write down the coefficients of the variables and the constants from each equation. The system of equations is: \(3x + 4y = 7\) \(4x - 2y = 5\)

- Form the Coefficient Matrix

Next, form the coefficient matrix. This matrix includes only the coefficients of the variables: \[ \begin{pmatrix} 3 & 4 \ 4 & -2 \ }\]

- Append the Constants

To form the augmented matrix, append the constants from each equation to the coefficient matrix as a new column: \[ \begin{pmatrix} 3 & 4 & \big| & 7 \ 4 & -2 & \big| & 5 \ }\]

Key concepts

System of Equations

A system of equations is a set of two or more equations that share common variables. In our example, we have the system:\[\begin{array}{l}3x + 4y = 7 \4x - 2y = 5\end{array}\]Here, both equations involve the variables 'x' and 'y'. The solution to the system is the set of values for 'x' and 'y' that satisfy both equations simultaneously.
This is a key concept in linear algebra and various other fields.
This means finding points where both lines intersect on a graph.
In our example, solving for 'x' and 'y' will give us the point where these two lines intersect.
Understanding this concept is critical for analyzing many real-world situations, such as solving problems related to physics, economics, and engineering.

Coefficient Matrix

The coefficient matrix is derived from the coefficients of the variables in a system of equations. This matrix does not include the constants from the equations, only the coefficients of the terms involving variables.
For our given system of equations:\[\begin{array}{l}3x + 4y = 7 \4x - 2y = 5\end{array}\]The coefficient matrix is:
\[\begin{pmatrix}3 & 4 \4 & -2\end{pmatrix}\]This matrix will help us perform matrix operations efficiently to solve the system of equations. It's vital to correctly identify the coefficients of each variable to construct the coefficient matrix accurately.
The matrix encapsulates essential information about the linear transformations associated with the system.
Mastering how to form and manipulate coefficient matrices is fundamental in linear algebra, as it serves as the starting point for more complex matrix operations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. Understanding linear algebra involves learning how to solve systems of linear equations using methods such as matrix operations, determinants, and vector spaces.
Key applications include:

• Computer Graphics: transforming and projecting images.
• Physics: solving mechanical systems and electrical circuits.
• Economics: modeling economic activities and optimizing solutions.
• Engineering: analyzing structures, signal processing, and more.

Augmented Matrix:
An augmented matrix merges the coefficient matrix with the constants from the equations. In our example:
\[\begin{pmatrix}3 & 4 & | & 7 \4 & -2 & | & 5\end{pmatrix}\]This augmented matrix represents the entire system in a format suitable for applying matrix operations to find the solution. Learning how to form and use augmented matrices is crucial for simplifying the process of solving systems of linear equations and an essential skill in linear algebra.