Problem 103 - Precalculus

Question

Explain the mistake that is made. Solve the system of equations using an augmented matrix. $$ \begin{array}{l} y-x+z=2 \\ x-2 z+y=-3 \\ x+y+z=6 \end{array} $$ Solution: Step 1: Write as an augmented matrix. \(\left[\begin{array}{rrr|r}1 & -1 & 1 & 2 \\ 1 & -2 & 1 & -3 \\ 1 & 1 & 1 & 6\end{array}\right]\) Step 2: Reduce the matrix using Gaussian elimination. \(\left[\begin{array}{rrr|r}1 & -1 & 1 & 2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & -6\end{array}\right]\) Step 3: Identify the solution. Row 3 is inconsistent, so there is no solution. This is incorrect. The correct answer is \(x=1, y=2, z=3\) What mistake was made?

Step-by-Step Solution

Answer

Incorrect initial row setup in matrix; row operation should yield consistent equations. Rearranging yields correct solution.

Step 1: Identify the Goal

We need to find the mistake in solving the given system of equations using an augmented matrix. We'll determine where the errors were made in the solution provided.

Step 2: Check the Augmented Matrix Setup

An augmented matrix is used to represent the system of linear equations. The initial matrix provided \(\begin{bmatrix} 1 & -1 & 1 & 2 \ 1 & -2 & 1 & -3 \ 1 & 1 & 1 & 6 \end{bmatrix}\) needs to correctly represent the equations given. We compare the coefficients and constants between the matrix and the equations.

Step 3: Verify Coefficient Arrangement

Check that each row in the matrix corresponds to an equation with the form: 1. For the first equation \(y - x + z = 2\), coordinates should be \([-1, 1, 1, 2]\).2. For the second equation \(x - 2z + y = -3\), coordinates should be \([1, 1, -2, -3]\).3. For the third equation \(x + y + z = 6\), coordinates should be \([1, 1, 1, 6]\).The matrix is initially incorrectly set up.

Step 4: Correct the Matrix Representation

Rearrange the initial matrix rows to correspond correctly to the equations \(\begin{bmatrix} -1 & 1 & 1 & 2 \ 1 & 1 & -2 & -3 \ 1 & 1 & 1 & 6 \end{bmatrix}\). Reorder the equations or their rows for consistency.

Step 5: Review Row Reduction Steps

Perform Gaussian elimination with the correct initial matrix. Each step should be reviewed from row swapping, scaling, and adding/subtracting multiples of rows from one another to reach upper-triangular form, while accurately targeting leading ones and zeros filling sub-diagonals.

Step 6: Compare with Correct Solution

Given that the expected correct solution is \(x=1, y=2, z=3\), ensure each matrix manipulation step leads towards this solution without inconsistencies appearing, as they did in the initial attempt with '0 = -6.' This confirms an error in row operations or initial setup was present.

Key Concepts

Gaussian Elimination

Gaussian Elimination is a popular method for solving systems of linear equations. The technique involves performing row operations on the augmented matrix of a system to bring it to a form that is easier to interpret. This is commonly known as row-echelon form.

The steps involved in Gaussian Elimination include:

• Swapping rows: This can be employed if it's more convenient for eliminating coefficients.
• Multiplying a row by a non-zero scalar: This is used to simplify numbers.
• Adding or subtracting a multiple of one row to another: This is the core of reducing a matrix.

By consistently applying these operations, you can transform the matrix to decode unknown variables more clearly. The end goal is usually to achieve a unique leading entry of 1 in every row, known as the leading 1, while ensuring zeros appear below this marker. These steps, when executed precisely, reveal the solutions of the system or may show the system's inconsistency.

System of Equations

A System of Equations involves two or more equations with multiple variables. The goal is to find the value(s) of the variables that satisfy all the equations simultaneously.

In many cases, systems of equations can be categorized into three types based on solutions:

• Consistent and Independent: One unique solution exists, where all equations intersect at a single point.
• Consistent and Dependent: Infinite solutions exist, typically when equations represent the same line or plane.
• Inconsistent: No solutions exist as the equations represent parallel lines or planes that never intersect.

Representing the system in an augmented matrix is a systematic approach. Each row corresponds to an equation, and each column represents the coefficients of variables. The column after the vertical bar in the matrix holds the constants from each equation. This conversion into matrix form is the first step before using methods such as Gaussian or Gauss-Jordan elimination.

Matrix Row Reduction

Matrix Row Reduction is the process of transforming a matrix into a simpler form. This often involves utilizing Gaussian or Gauss-Jordan elimination techniques. The goal is to reach row-echelon form or reduced row-echelon form.

With row reduction, the following crucial moves are key:

• Eliminating leading coefficients below the pivot (leading 1) for each row: Ensures zeros fill sub-diagonal spaces.
• Focusing on sequentially "clearing out" one column at a time, moving through each row: This directs actions toward simplifying and solving.
• Reaching a state where back-substitution can reveal variables easily if needed.

In the example provided with the system of equations, the mistake occurred during the incorrect row reduction steps. Following the proper execution of these row operations achieves the correct solution \(x=1, y=2, z=3\). This indicates that earlier errors led to the incorrect conclusion of the solution being inconsistent.