Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{l} 2 x+3 y-6=0 \\ 4 x-6 y+2=0 \end{array}\right. $$

Short answer

[[2, 3, -6], [4, -6, 2]]

Step-by-step solution

Write the coefficients

Identify the coefficients of the variables and the constants in each of the equations. For the equation \(2x + 3y - 6 = 0\), the coefficients are 2, 3 and the constant is -6. For the equation \(4x - 6y + 2 = 0\), the coefficients are 4, -6 and the constant is 2.

Set up the augmented matrix

Create the augmented matrix using the coefficients and constants identified in Step 1. The augmented matrix for the system is arranged as follows: \[ \begin{bmatrix} 2 & 3 & | & -6 \ 4 & -6 & | & 2 \end{bmatrix} \] where the vertical line represents the separation between the coefficients of the variables and the constants.

Key concepts

Systems of Equations

A system of equations is a set of two or more equations with the same variables. For instance, in the given exercise, we have two equations involving the variables \(x\) and \(y\). The goal is to find the values of these variables that satisfy all the equations in the system simultaneously.
There are different methods to solve systems of equations, such as substitution, elimination, and matrix methods.
Understanding how to represent and solve such systems is crucial as it lays the foundation for more advanced topics in mathematics and its applications.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations.
It's a powerful tool used in various fields like engineering, computer science, physics, economics, and more. In this exercise, we focus on a fundamental concept within linear algebra: using matrices to represent and solve systems of linear equations.
By converting the system into an augmented matrix, we can use matrix operations to find the solution efficiently.

Matrix Representation

Matrix representation involves expressing a system of linear equations as a matrix. This makes it easier to manipulate and solve the system.
In an augmented matrix, each row corresponds to one equation of the system, and each column represents the coefficients of one variable, with an additional column for constants.
For example, the system given in the exercise:
\[ \begin{cases} 2x + 3y = 6 \ 4x - 6y = -2 \ \end{cases} \]
is represented as:
\[ \begin{bmatrix} 2 & 3 & | & -6 \ 4 & -6 & | & 2 \end{bmatrix} \]
Here, the vertical line divides the coefficients of the variables from the constants, making it easier to perform elimination.

Coefficients

Coefficients are the numerical factors that multiply the variables in an equation.
In the given exercise, for the equation \(2x + 3y - 6 = 0\), the coefficients of \(x\) and \(y\) are 2 and 3, respectively. Similarly, for \(4x - 6y + 2 = 0\), the coefficients are 4 for \(x\) and -6 for \(y\).
Correctly identifying these coefficients is essential when setting up your augmented matrix.
You line them up in rows and columns to help you apply matrix operations accurately.

Constants

Constants in a system of linear equations are the standalone numbers without variables.
In our exercise problem, the constants in the equations are -6 in the first equation \(2x + 3y - 6 = 0\) and 2 in the second equation \(4x - 6y + 2 = 0\).
When creating the augmented matrix, these constants are placed in the rightmost column. The representation helps to visualize and perform operations like Gaussian elimination to solve the system.
Getting a grip on constants enhances understanding of the relationships between variables in the equations.