Question

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

Short answer

The augmented matrix is \begin{bmatrix} 5 & -1 & -1 & | & 0 \ 1 & 1 & 0 & | & 5 \ 2 & 0 & -3 & | & 2 \end{bmatrix}.

Step-by-step solution

- Identify the coefficients

First, identify the coefficients of each variable in each equation. Organize the coefficients and constants separately.

- Write the first row

For the first equation, extract the coefficients of each variable and the constant: 5, -1, -1 for x, y, z, and 0 as the constant.

- Write the second row

For the second equation, extract the coefficients: 1 for x, 1 for y, 0 for z (since z is not present), and 5 as the constant.

- Write the third row

For the third equation, extract the coefficients: 2 for x, 0 for y (since y is not present), -3 for z, and 2 as the constant.

- Form the augmented matrix

Combine all the rows from steps 2 to 4 to form the augmented matrix. Place the constants in a separate column to the right.

Key concepts

system of equations

A system of equations is a collection of two or more equations with the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously.

Systems of equations can be solved using various methods, such as substitution, elimination, and matrix methods.

In this example, we are given three equations with the variables x, y, and z that are connected:

• 5x - y - z = 0
• x + y = 5
• 2x - 3z = 2

To solve this system of equations efficiently, we can use matrix methods, which involve organizing the coefficients of the equations into a matrix and then solving it.

matrix formation

Matrix formation is a technique used to represent a system of equations in a compact and organized way. This is particularly useful in linear algebra.

Steps to form an augmented matrix:

• Identify the coefficients: Each equation has coefficients for its variables and a constant term.
• Write each row: The coefficients for the variables and their constant terms form the rows.
• Combine the rows: These rows are combined to form the augmented matrix.

For the given system of equations:

• The first row represents 5x - y - z = 0, with coefficients 5, -1, -1, and constant 0.
• The second row represents x + y = 5, with coefficients 1, 1, 0 (for z), and constant 5.
• The third row represents 2x - 3z = 2, with coefficients 2, 0 (for y), -3, and constant 2.

Thus, the augmented matrix is:
\[\begin{bmatrix} 5 & -1 & -1 & | & 0 \ 1 & 1 & 0 & | & 5 \ 2 & 0 & -3 & | & 2 \end{bmatrix}\]

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), and linear transformations between these spaces.

This field is foundational for many areas of mathematics and engineering, including computer graphics, optimization, and machine learning.

In linear algebra, systems of linear equations can be solved using matrices and matrix operations. By representing a system of equations as an augmented matrix, we can use methods like Gaussian elimination and matrix inversion to find solutions.

In Gaussian elimination, we manipulate the matrix to achieve a form where the solutions can be easily interpreted.

Understand that forming and solving augmented matrices gives us a powerful toolkit for working with linear systems.