Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-2 y+3 z=6 \\ 4 x-3 y+2 z=0 \\ -2 x+3 y-7 z=1 \end{array}\right. $$

Short answer

The system is inconsistent.

Step-by-step solution

- Write the augmented matrix

Convert the given system of equations into an augmented matrix.\[ \begin{bmatrix} 2 & -2 & 3 & | & 6 \ 4 & -3 & 2 & | & 0 \ -2 & 3 & -7 & | & 1 \end{bmatrix} \]

- Perform Row 1 operations

Divide the first row by 2 to get a leading 1 in the first column.\[ \begin{bmatrix} 1 & -1 & 1.5 & | & 3 \ 4 & -3 & 2 & | & 0 \ -2 & 3 & -7 & | & 1 \end{bmatrix} \]

- Eliminate below pivot (Row 2)

Eliminate the first column entry for the second row by replacing Row 2 with \(R_2 - 4R_1\).\[ \begin{bmatrix} 1 & -1 & 1.5 & | & 3 \ 0 & 1 & -4 & | & -12 \ -2 & 3 & -7 & | & 1 \end{bmatrix} \]

- Eliminate below pivot (Row 3)

Eliminate the first column entry for the third row by replacing Row 3 with \(R_3 + 2R_1\).\[ \begin{bmatrix} 1 & -1 & 1.5 & | & 3 \ 0 & 1 & -4 & | & -12 \ 0 & 1 & -4 & | & 7 \end{bmatrix} \]

- Eliminate duplicate row

Notice that Rows 2 and 3 are essentially the same. Subtract Row 2 from Row 3.\[ \begin{bmatrix} 1 & -1 & 1.5 & | & 3 \ 0 & 1 & -4 & | & -12 \ 0 & 0 & 0 & | & 19 \end{bmatrix} \]

- Identify inconsistency

The last row translates to \(0x + 0y + 0z = 19\), which is impossible. Hence, the system is inconsistent.

Key concepts

Solving Systems of Equations

Solving systems of equations involving multiple variables is an essential skill in algebra. There are various methods to achieve this, like substitution, elimination, and matrix row operations. Today, we'll focus on using matrices and row operations to solve a system of linear equations. The goal is to manipulate the matrix into reduced row echelon form (RREF) to easily identify the solutions.
We convert the system into an augmented matrix and perform row operations to simplify it. Row operations include:

• Swapping rows.
• Multiplying a row by a nonzero constant.
• Adding or subtracting multiples of rows from each other.

By applying these operations, we aim to create zeros below the leading 1s (also called pivots). If we can reduce the matrix straightforwardly, we can often read off the solutions directly from it. Let's dive deeper into augmented matrices and what they represent.

Augmented Matrices

An augmented matrix is a compact way to represent a system of equations. It includes both the coefficients of the variables and the constants on the right-hand side of the equations. These matrices are very useful because they allow us to apply row operations systematically.
For example, consider the system of equations:

\( 2x - 2y + 3z = 6 \)
\( 4x - 3y + 2z = 0 \)
\( -2x + 3y - 7z = 1 \)
This system can be written as the augmented matrix:
\( \begin{bmatrix} 2 & -2 & 3 & | & 6 \ 4 & -3 & 2 & | & 0 \ -2 & 3 & -7 & | & 1 \ \end{bmatrix} \)
The vertical bar separates the coefficients of the variables from the constants. To solve the system, we use row operations to reach a form where the solutions become obvious. The goal is to get a leading 1 in every row, with zeros below each leading 1, and ideally, zeros above them too. This form is known as row echelon form. Now, let's discuss what happens when a system of equations has no solution.

Inconsistent Systems

An inconsistent system occurs when there are no solutions that satisfy all the equations simultaneously. This is often discovered during the process of row reduction.
In our example, after performing a series of row operations, we ended up with the following matrix:
\( \begin{bmatrix} 1 & -1 & 1.5 & | & 3 \ 0 & 1 & -4 & | & -12 \ 0 & 0 & 0 & | & 19 \ \end{bmatrix} \)
The last row of this matrix says \(0x + 0y + 0z = 19 \), which translates to \( 0 = 19 \). This is clearly a contradiction. Such a row, where all variable coefficients are zero but the constant is non-zero, indicates that there is no combination of values for x, y, and z that can satisfy all the given equations.
Hence, we conclude that the system is inconsistent, meaning there are no solutions. Identifying inconsistencies is crucial. It helps avoid futile efforts in solving and redirects focus to correct representation or formulation of the system.