Question

Write the augmented matrix of the given system of equations. $$ \left\\{\begin{array}{r} x-y+z=10 \\ 3 x+3 y=5 \\ x+y+2 z=2 \end{array}\right. $$

Short answer

[ \[ \begin{bmatrix} 1 & -1 & 1 & | & 10 \ 3 & 3 & 0 & | & 5 \ 1 & 1 & 2 & | & 2 \end{bmatrix} \] ]

Step-by-step solution

Identify the system of equations

The given system of equations is: 1. \(x - y + z = 10\) 2. \(3x + 3y = 5\) 3. \(x + y + 2z = 2\)

Write the coefficient matrix

Extract the coefficients of the variables (x, y, z) from each equation and write them in a matrix form. This gives: \[ \begin{bmatrix} 1 & -1 & 1 \ 3 & 3 & 0 \ 1 & 1 & 2 \end{bmatrix} \]

Write the constants

Extract the constants from each equation and write them as a column matrix. This gives: \[ \begin{bmatrix} 10 \ 5 \ 2 \end{bmatrix} \]

Form the augmented matrix

Combine the coefficient matrix and the constants column matrix to form the augmented matrix. This is done by placing the constants column matrix to the right of the coefficient matrix separated by a vertical line: \[ \begin{bmatrix} 1 & -1 & 1 & | & 10 \ 3 & 3 & 0 & | & 5 \ 1 & 1 & 2 & | & 2 \end{bmatrix} \]

Key concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. These equations are to be solved simultaneously, meaning we need to find a common solution that satisfies all equations in the system. For instance, in the equation:
\( x - y + z = 10 \),
\( 3x + 3y = 5 \),
\( x + y + 2z = 2 \),
we have a system of three equations with three variables (x, y, and z).
Understanding how to solve such systems is crucial for many applications in science, engineering, and beyond.

Coefficient Matrix

The coefficient matrix is formed by extracting the coefficients of the variables from the given system of equations.
In our example, for the system:
\( x - y + z = 10 \),
\( 3x + 3y = 5 \),
\( x + y + 2z = 2 \),
the coefficient matrix is:
\[\begin{bmatrix} 1 & -1 & 1 \ 3 & 3 & 0 \ 1 & 1 & 2 \end{bmatrix} \].
This matrix provides a compact way of representing the linear relationships among the variables.

Matrix Operations

Matrix operations are fundamental tools in linear algebra, including addition, subtraction, multiplication, and finding the inverse. In solving systems of equations, we often employ matrix operations to simplify and solve the matrices involved.
Common operations used include:

• Row Echelon Form (REF)
• Gaussian elimination
• Matrix multiplication

.
These operations help us manipulate the matrices to find the solution to the system of equations.

Column Matrix

A column matrix, also known as a column vector, is a matrix with a single column. In the context of systems of equations, the column matrix represents the constants on the right-hand side of the equations.
For example, in the system:
\( x - y + z = 10 \),
\( 3x + 3y = 5 \),
\( x + y + 2z = 2 \),
the corresponding column matrix of constants is:
\[\begin{bmatrix} 10 \ 5 \ 2 \end{bmatrix} \].
This matrix is combined with the coefficient matrix to form the augmented matrix.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is a foundational subject in mathematics with broad applications in natural sciences, engineering, computer science, economics, and more.
In linear algebra, we study systems of linear equations, vector spaces, and matrix transformations.
The example system of equations and its corresponding augmented matrix demonstrate the practical application of linear algebra concepts in solving real-world problems.