Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{rr} 2 x+3 y-4 z= & 0 \\ x-5 z+2= & 0 \\ x+2 y-3 z= & -2 \end{array}\right. $$

Short answer

\[ \begin{pmatrix} 2 & 3 & -4 & | & 0 \ 1 & 0 & -5 & | & -2 \ 1 & 2 & -3 & | & -2 \end{pmatrix}\]

Step-by-step solution

- Identify the coefficients of each variable

Look at each equation and identify the coefficients of the variables x, y, and z. Also, identify the constants on the right side of the equation. For the system given: 1. Equation: \[2x + 3y - 4z = 0\] Coefficients: (2, 3, -4), Constant: 0 2. Equation: \[x - 5z + 2 = 0\] Coefficients: (1, 0, -5), Constant: -2 3. Equation: \[x + 2y - 3z = -2\] Coefficients: (1, 2, -3), Constant: -2

- Write the coefficients in matrix form

Create a matrix with the coefficients from each variable in order. Append the constants as the last column to form the augmented matrix. Taking the coefficients and constants identified: First row: (2, 3, -4 | 0) Second row: (1, 0, -5 | -2) Third row: (1, 2, -3 | -2)

- Format the augmented matrix

Combine the rows to write the augmented matrix: \[ \begin{pmatrix} 2 & 3 & -4 & | & 0 \ 1 & 0 & -5 & | & -2 \ 1 & 2 & -3 & | & -2 \end{pmatrix}\]

Key concepts

system of linear equations

A system of linear equations is a set of two or more linear equations involving the same set of variables. Each equation represents a straight line, and the solution to the system is the point or points where these lines intersect. For example, in the given exercise, we have three equations involving the variables x, y, and z: 2x + 3y - 4z = 0, x - 5z + 2 = 0, and x + 2y - 3z = -2. Solving this system means finding the values of x, y, and z that satisfy all three equations simultaneously. This is often done using methods such as substitution, elimination, or matrix techniques.

matrix representation

Matrix representation is a way of organizing the coefficients and constants of a system of linear equations into a compact, structured form. This makes it easier to manipulate and solve the system using linear algebra techniques. In the matrix form, each equation's coefficients are arranged in rows, and each variable's coefficients form columns. This structured approach provides a visual and computational advantage. For example, the system of equations in the exercise is represented in matrix form as follows:

• First row: (2, 3, -4 | 0)
• Second row: (1, 0, -5 | -2)
• Third row: (1, 2, -3 | -2)

. This matrix showcases the organized arrangement of all coefficients and constants from the equations.

coefficients of variables

The coefficients of variables in a system of linear equations are the numerical factors that multiply the variables in each equation. Identifying these coefficients is crucial for forming the augmented matrix. For our exercise, the coefficients are as follows:
1. For the equation 2x + 3y - 4z = 0, the coefficients are (2, 3, -4).
2. For the equation x - 5z + 2 = 0, the coefficients are (1, 0, -5).
3. For the equation x + 2y - 3z = -2, the coefficients are (1, 2, -3).
By extracting these coefficients, we format them into a matrix to streamline the process of solving the equations.

constant terms

Constant terms in a system of linear equations are the standalone numbers on the right side of the equation, not multiplied by any variable. These constants shift the lines represented by the equations up or down in the graph. For the system provided in the exercise, the constants are:
1. For the equation 2x + 3y - 4z = 0, the constant is 0.
2. For the equation x - 5z + 2 = 0, the constant is -2.
3. For the equation x + 2y - 3z = -2, the constant is -2.
Including these constants in the augmented matrix helps provide the complete picture of the system of equations, aiding in the computational process to find the solution.