Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{r} x-y=6 \\ 2 x-3 z=16 \\ 2 y+z=4 \end{array}\right. $$

Short answer

The solution is \(x = 8, y = 2, z = 0\).

Step-by-step solution

Write the system of equations as an augmented matrix

Convert the system of linear equations into an augmented matrix. The system is: \(x - y = 6 \) \(2x - 3z = 16\) \(2y + z = 4\)The augmented matrix will be: \[\begin{pmatrix}1 & -1 & 0 & | & 6\ 2 & 0 & -3 & | & 16\ 0 & 2 & 1 & | & 4\begin{pmatrix}\]

Eliminate the first element of the second row

Subtract 2 times the first row from the second row to eliminate the '2' in the second row, first column: \( R_2 \rightarrow R_2 - 2R_1 \) The matrix becomes: \[\begin{pmatrix}1 & -1 & 0 & | & 6\ 0 & 2 & -3 & | & 4\ 0 & 2 & 1 & | & 4\begin{pmatrix}\]

Eliminate the second row's second element from the third row

Subtract the second row from the third row to eliminate the '2' in the third row's second column: \( R_3 \rightarrow R_3 - R_2 \) The matrix becomes: \[\begin{pmatrix}1 & -1 & 0 & | & 6\ 0 & 2 & -3 & | & 4\ 0 & 0 & 4 & | & 0\begin{pmatrix}\]

Solve for the variables

From the third row, we get: \(4z = 0 \Rightarrow z = 0\)Substitute \(z = 0\) into the second row: \(2y - 3(0) = 4 \Rightarrow 2y = 4 \Rightarrow y = 2\)Substitute \(y = 2\) into the first row: \(x - 2 = 6 \Rightarrow x = 8\)Thus, the solution to the system is: \(x = 8, y = 2, z = 0\).

Key concepts

row operations

Row operations are essential actions in linear algebra used to simplify matrices and solve systems of equations. These operations help transform matrices into simpler forms, like row echelon form or reduced row echelon form.
The three main types of row operations are:

• Swapping two rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting a multiple of one row to/from another row.

Each of these operations is performed to manipulate the matrix without changing the solutions to the system of equations.
For instance, in the problem provided, subtracting 2 times the first row from the second row (denoted as \( R_2 \rightarrow R_2 - 2R_1 \)) simplifies the matrix and helps eliminate specific terms, making the system easier to solve.

augmented matrix

An augmented matrix is a way to represent a system of linear equations. It combines the coefficient matrix with the constants from each equation into one matrix, separating them typically by a vertical line.
For the given system of equations:
\( x - y = 6 \) \( 2x - 3z = 16 \) \( 2y + z = 4 \)
The augmented matrix form is:
\[ \begin{pmatrix} 1 & -1 & 0 & | & 6 \ 2 & 0 & -3 & | & 16 \ 0 & 2 & 1 & | & 4 \end{pmatrix} \]
This representation can then be used to apply row operations systematically to find the solution to the system. The augmented matrix concisely encapsulates all the necessary information for solving the equations using matrices.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. When solving systems of equations, linear algebra provides the tools and methods needed to find solutions using matrix operations.
Techniques such as Gaussian elimination and matrix transformations are cornerstones of linear algebra.
In our example, we used linear algebra methods to convert a system of linear equations into an augmented matrix, which was then systematically simplified using row operations. This process eventually leads to the solution of the equations.
Understanding concepts like row operations and augmented matrices within linear algebra allows you to tackle a wide range of problems in both pure and applied mathematics efficiently.