Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 2 x+y-3 z &=1 \\ x+y+2 z &=5 \\ 3 x+2 y-z &=-3 \end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution, as the matrices show a contradiction.

Step 1: Write the Augmented Matrix

First, convert the system of equations into an augmented matrix. The coefficients of the variables and the constants form the matrix:\[\begin{bmatrix}2 & 1 & -3 & | & 1 \1 & 1 & 2 & | & 5 \3 & 2 & -1 & | & -3\end{bmatrix}\]

Step 2: Make Pivot in First Column

Use row operations to make the first element of the first column into a leading 1. Swap Row 1 and Row 2 to simplify the process:\[\begin{bmatrix}1 & 1 & 2 & | & 5 \2 & 1 & -3 & | & 1 \3 & 2 & -1 & | & -3\end{bmatrix}\]

Step 3: Eliminate First Column Below Pivot

Eliminate the elements below the pivot in the first column. Subtract 2 times Row 1 from Row 2, and 3 times Row 1 from Row 3:\[R_2 = R_2 - 2R_1 \R_3 = R_3 - 3R_1\]This results in:\[\begin{bmatrix}1 & 1 & 2 & | & 5 \0 & -1 & -7 & | & -9 \0 & -1 & -7 & | & -18\end{bmatrix}\]

Step 4: Simplify Second Row

Divide the entire Row 2 by -1 to get a leading 1:\[R_2 = -R_2 \\begin{bmatrix}1 & 1 & 2 & | & 5 \0 & 1 & 7 & | & 9 \0 & -1 & -7 & | & -18\end{bmatrix}\]

Step 5: Eliminate Second Column Below Pivot

Add Row 2 to Row 3 to eliminate the element below the pivot in the second column:\[R_3 = R_3 + R_2 \ \begin{bmatrix}1 & 1 & 2 & | & 5 \0 & 1 & 7 & | & 9 \0 & 0 & 0 & | & -9\end{bmatrix}\]This reveals a contradiction as the third row implies an inconsistent statement "0=9", indicating no solution.

Key Concepts

Row Operations

Row operations are the key techniques used to manipulate matrices for solving systems of equations. They help simplify the matrix while keeping the system's solutions the same. These operations include three main types: exchanging two rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. These operations make it possible to transform the matrix into a simpler form, typically row-echelon or reduced row-echelon form, facilitating the solving process.
For instance, in the provided solution, such operations are crucial in forming pivot elements (ones) in successive columns, under which the elements are zeros. This structured approach helps isolate variables step by step. By starting with the first column and working your way across to other columns, row operations guide the matrix towards a state where the values of each variable in the system can be extracted easily.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. Solving a system means finding the values for the variables that satisfy all equations simultaneously. In mathematical notation, a typical system might look like:

• Equation 1: \( ax + by = c \)
• Equation 2: \( dx + ey = f \)
• And so on...

These systems can be solved using various methods, like substitution, elimination, or using matrices. The matrix method employs an augmented matrix to represent the system, which streamlines operations on the equations.
A key point in using matrices to solve these systems is reducing the augmented matrix through row operations, aiming for a clear path to identify the variable values. The ultimate goal is to rewrite the system in a simple form, like triangular or diagonal, where each row corresponds distinctly to each variable, making it easier to solve the equations step by step.

Inconsistent System

An inconsistent system is one where no solution exists that satisfies all equations in the system simultaneously. This typically occurs when, after performing all necessary row operations and simplifying the matrix, you end up with a contradiction. A contradiction might look something like this: \( 0 = 9 \).
In the given problem, the final row of the matrix implied such a contradiction. This insight tells us that no combination of numbers will satisfy all the given equations together. When visualized on a graph, inconsistent systems often reflect parallel lines that never intersect, indicating that no common solutions exist.
Recognizing inconsistency is crucial as it saves time and effort. It is important to be attentive to any signs of contradictions while performing row operations and interpreting matrix results, thus ensuring an efficient solution process.