Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{r} 9 x-y=0 \\ 3 x-y-4=0 \end{array}\right. $$

Short answer

The augmented matrix is: \( \begin{pmatrix} 9 & -1 & 0 \ 3 & -1 & 4 \end{pmatrix} \).

Step-by-step solution

Write down the coefficients of each variable

Identify the coefficients of each variable in the given system. For the first equation, the coefficients are 9 (for x) and -1 (for y). For the second equation, the coefficients are 3 (for x) and -1 (for y).

Identify the constants

Locate the constants in the equations. For the first equation, the constant is 0. For the second equation, rearrange the equation to isolate the constant: $$ 3x - y = 4 $$ by adding 4 to both sides, the constant is 4.

Formulate the augmented matrix

Place the coefficients and constants in matrix form. The augmented matrix includes the coefficients of x and y followed by the constants. Therefore, the matrix is: $$ \begin{pmatrix} 9 & -1 & 0 \ 3 & -1 & 4 \end{pmatrix} $$.

Key concepts

Systems of Equations

A system of equations consists of multiple equations that share common variables. These equations are usually solved simultaneously to find values for these variables that satisfy all the equations at the same time. For example, in our given system:
\[ 9x - y = 0 \]
\[ 3x - y = 4 \]
We have two equations with two variables, x and y. The goal is to find values for x and y that make both equations true. Solving systems of equations can be done using various methods, such as substitution, elimination, or using matrices.

Matrix Representation

Matrix representation is a convenient way to organize and solve systems of linear equations. Instead of dealing with multiple equations with variables, we can represent these equations in a compact matrix form.
To do this, we'll write the coefficients of each variable and the constants as an array. This array (matrix) helps simplify calculations and can be easily manipulated using matrix operations.
In the given system: \[ 9x - y = 0 \] \[ 3x - y = 4 \]
The matrix representation will include the coefficients of x and y, and the constants from each equation. For example, the augmented matrix form is: \[ \begin{pmatrix} 9 & -1 & 0 \ 3 & -1 & 4 \end{pmatrix} \]
This matrix includes the coefficient of x in the first column, the coefficient of y in the second column, and the constants in the third column. It allows us to apply matrix techniques to solve the system.

Coefficients and Constants

In any system of linear equations, each term consists of a coefficient multiplied by a variable. The coefficients are the numerical factors in front of the variables, and the constants are the standalone numbers that are not accompanied by variables.
For example, in the first equation of our system: \[ 9x - y = 0 \]
The coefficient for x is 9, and for y, it is -1. The constant term on the right side of the equation is 0.
In the second equation: \[ 3x - y = 4 \]
Here, the coefficient for x is 3, and for y, it is -1. The constant is 4.
Identifying these coefficients and constants accurately is essential to forming the augmented matrix, which allows us to solve the equations. By understanding the roles of coefficients and constants, you can easily translate a system of equations into a matrix form and use matrix operations to find solutions.