Problem 23 - Precalculus: Mathematical for Calculus

Question

Find the complete solution of the linear system, or show that it is inconsistent. $$\left\\{\begin{array}{rr} x+2 y-z= & 1 \\ 2 x+3 y-4 z= & -3 \\ 3 x+6 y-3 z= & 4 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent, as indicated by a row leading to the equation \(0 = 1\).

Step 1: Convert System to Augmented Matrix

First, write the system of equations as an augmented matrix. The matrix will represent the coefficients of the variables and the constants from the right side of the equations:\[\begin{bmatrix}1 & 2 & -1 & | & 1 \2 & 3 & -4 & | & -3 \3 & 6 & -3 & | & 4\end{bmatrix}\]

Step 2: Use Row Operations to Achieve Row Echelon Form

Perform row operations to transform the matrix into row echelon form. To eliminate the variable \(x\) from the second and third rows:- Replace Row 2 with (Row 2 - 2 * Row 1): \[0 \, 1 \, -2 \, | -5\]- Replace Row 3 with (Row 3 - 3 * Row 1): \[0 \, 0 \, 0 \, | 1\]The matrix now is:\[\begin{bmatrix}1 & 2 & -1 & | & 1 \0 & 1 & -2 & | & -5 \0 & 0 & 0 & | & 1\end{bmatrix}\]

Step 3: Check for Consistency

Observe the last row: \(0x + 0y + 0z = 1\), which simplifies to \(0 = 1\). This statement is a contradiction, meaning the system of equations has no solution.

Key Concepts

Augmented Matrix

To solve a linear system using matrices, we start by writing the system's equations as an augmented matrix. This type of matrix represents both the coefficients of the variables and the constants on the right side of each equation, all arranged in rows.

• The column before the vertical bar contains the coefficients of variables (in order).
• The column after the vertical bar contains the constants from the equations.

For instance, if we have a system of equations:\[\begin{align*}x + 2y - z &= 1 \2x + 3y - 4z &= -3 \3x + 6y - 3z &= 4\end{align*}\]The augmented matrix is:\[\begin{bmatrix}1 & 2 & -1 & | & 1 \2 & 3 & -4 & | & -3 \3 & 6 & -3 & | & 4\end{bmatrix}\]Thus, the columns effectively mirror the structure of the system, helping us guide through further transformations.

Row Operations

Once the system is represented as an augmented matrix, the next step involves using row operations to simplify it into a more manageable form. Row operations are elementary procedures applied to the rows of a matrix:

• Row Swapping: Interchanging two rows.
• Row Multiplication: Multiplying a row by a non-zero scalar.
• Row Addition: Adding or subtracting the multiple of one row to another.

In our exercise:

• Replace the second row with (Row 2 - 2 * Row 1).
• Replace the third row with (Row 3 - 3 * Row 1).

This process helps in reducing the matrix to a simpler form, ultimately leading towards solving for variables.

Row Echelon Form

The goal is to convert the augmented matrix into a row echelon form (REF). This is a transformation where each leading entry in a row is to the right of the leading entry in the row above.

• All non-zero rows are above any rows of all zeros.
• The leading entry in each non-zero row is 1 and located to the right of the leading entry in the previous row.
• All entries in a column below a leading entry are zeros.

This form simplifies understanding of the system's solutions, whether they exist, are unique or infinite. In our specific problem example, after performing necessary row operations,:\[\begin{bmatrix}1 & 2 & -1 & | & 1 \0 & 1 & -2 & | & -5 \0 & 0 & 0 & | & 1\end{bmatrix}\]If correctly done, this method shows either a clear path to solutions or contradictions.

Consistency of Equations

Finally, checking the consistency of the equations ensures whether the system has a solution or not. When evaluating row echelon form:

• An inconsistent system has a row that constructs a statement like \(0 = c\), where \(c\) is non-zero.
• A consistent system will allow for solving the variables without contradictions.

In our journey, we noticed that the third row of the matrix forms the equation \(0x + 0y + 0z = 1\). This derives the contradicting statement \(0 = 1\), labeling the system as inconsistent.
This means there is no possible combination of variables \(x, y, \) and \(z\) that satisfy all the equations. Understanding this helps in determining the feasibility of a given linear system.