Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-2 y-2 z=2 \\ 2 x+3 y+z=2 \\ 3 x+2 y=0 \end{array}\right. $$

Short answer

Inconsistent

Step-by-step solution

Write the Augmented Matrix

First, express the system of equations as an augmented matrix: \[ \begin{bmatrix} 2 & -2 & -2 & | & 2 \ 2 & 3 & 1 & | & 2 \ 3 & 2 & 0 & | & 0 \ \end{bmatrix} \]

Make the First Pivot

Ensure the first element (pivot) of the first column is 1. Divide the first row by 2: \[ \begin{bmatrix} 1 & -1 & -1 & | & 1 \ 2 & 3 & 1 & | & 2 \ 3 & 2 & 0 & | & 0 \ \end{bmatrix} \]

Eliminate First Column Below Pivot

Subtract 2 times the first row from the second row and subtract 3 times the first row from the third row: \[ \begin{bmatrix} 1 & -1 & -1 & | & 1 \ 0 & 5 & 3 & | & 0 \ 0 & 5 & 3 & | & -3 \ \end{bmatrix} \]

Make the Second Pivot

Ensure the second element (pivot) of the second column is 1. Divide the second row by 5: \[ \begin{bmatrix} 1 & -1 & -1 & | & 1 \ 0 & 1 & 0.6 & | & 0 \ 0 & 5 & 3 & | & -3 \ \end{bmatrix} \]

Eliminate Second Column Below Pivot

Subtract 5 times the second row from the third row: \[ \begin{bmatrix} 1 & -1 & -1 & | & 1 \ 0 & 1 & 0.6 & | & 0 \ 0 & 0 & 0 & | & -3 \ \end{bmatrix} \]

Check for Consistency

The third row simplifies to 0 = -3, which is a contradiction; hence, the system is inconsistent.

Key concepts

augmented matrix

To solve a system of equations, we often use matrices, particularly the augmented matrix. An augmented matrix includes the coefficients of the variables and the constants from each equation in a single matrix.
This helps in applying row operations to simplify the system. An augmented matrix is written in the form: \[ \left[ \begin{matrix} a & b & c & | & d \ e & f & g & | & h \ i & j & k & | & l \end{matrix} \right] \] Here, the vertical bar separates the coefficients of the variables from the constants.
In our exercise, for the system of equations given, the augmented matrix is: \[ \left[ \begin{matrix} 2 & -2 & -2 & | & 2 \ 2 & 3 & 1 & | & 2 \ 3 & 2 & 0 & | & 0 \end{matrix} \right] \]
This structure will allow us to perform row operations clearly and systematically.

system of equations

A system of equations is a set of two or more equations with the same variables. Solutions to such systems are values of the variables that satisfy all the equations simultaneously.
There are several methods to solve a system of equations, including substitution, elimination, and matrix methods using row operations. Using matrices can be particularly efficient for larger systems. In our example, we are given this system:
\[ \left\{ \begin{array}{r} 2x - 2y - 2z = 2 \ 2x + 3y + z = 2 \ 3x + 2y = 0 \end{array} \right. \] By expressing this system in augmented matrix form, we facilitate the use of row operations to solve it or determine its nature (e.g., consistent or inconsistent).

inconsistent system

An inconsistent system of equations is one that has no solutions. This occurs when there is a contradiction within the system.
A common sign of inconsistency in the row-reduced form of the augmented matrix is a row that translates to an impossible statement, such as 0 = -3. In our exercise:
\[ \left[ \begin{matrix} 1 & -1 & -1 & | & 1 \ 0 & 1 & 0.6 & | & 0 \ 0 & 0 & 0 & | & -3 \end{matrix} \right] \] The third row simplifies to 0 = -3, which is a contradiction. Hence, the given system is inconsistent, meaning there are no solutions that satisfy all three equations simultaneously.

pivot element

A pivot element in a matrix is the first non-zero element in a row, typically used in row operations to simplify a system of equations. It is crucial in transforming a matrix into its row echelon form (REF) or reduced row echelon form (RREF).
For example, in our exercise, the first pivot element is '2' in the first row, first column. To make working easier, we transform it to '1' by dividing the entire row by 2:
\[ \left[ \begin{matrix} 1 & -1 & -1 & | & 1 \ 2 & 3 & 1 & | & 2 \ 3 & 2 & 0 & | & 0 \end{matrix} \right] \]
Subsequent steps involve using the pivot to eliminate other elements in its column. By continually selecting pivot elements and performing row operations, we systematically simplify the matrix to determine solutions or identify inconsistencies.