Problem 32 - Precalculus: Mathematical for Calculus

Question

Find the complete solution of the linear system, or show that it is inconsistent. $$\left\\{\begin{array}{r} x+y+z+w=0 \\ x+y+2 z+2 w=0 \\ 2 x+2 y+3 z+4 w=1 \\ 2 x+3 y+4 z+5 w=2 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent and has no solution.

Step 1: Write the system in matrix form

First, we express the system of linear equations in the form of a matrix.The system can be written as:\[\begin{bmatrix} 1 & 1 & 1 & 1 \1 & 1 & 2 & 2 \2 & 2 & 3 & 4 \2 & 3 & 4 & 5 \end{bmatrix}\begin{bmatrix} x \y \z \w \end{bmatrix} = \begin{bmatrix} 0 \0 \1 \2 \end{bmatrix}.\]

Step 2: Row-reduce the matrix

Perform row operations to bring the augmented matrix to its reduced row-echelon form:1\. Subtract the first row from the second row.2\. Subtract twice the first row from the third row.3\. Subtract twice the first row from the fourth row.The matrix becomes:\[\begin{bmatrix}1 & 1 & 1 & 1 & 0 \0 & 0 & 1 & 1 & 0 \0 & 0 & 1 & 2 & 1 \0 & 1 & 2 & 3 & 2\end{bmatrix}.\]

Step 3: Continue row reduction

Continue reducing the matrix:1\. Subtract the second row from the third row.2\. Subtract twice the second row from the fourth row.The matrix now looks like this:\[\begin{bmatrix} 1 & 1 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 1 & 1 \ 0 & 1 & 0 & 1 & 2 \end{bmatrix}.\]

Step 4: Identify contradictions

Look for any rows that indicate contradictions in the system:Since the third row implies \(0 = 1\), this indicates a contradiction, showing that there is no solution.This means the system of equations is inconsistent.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. Each equation represents a straight line in a coordinate space, and the solution to the system is the point or points where these lines intersect. In our case, we have four equations in four variables:

• Equation 1: \(x + y + z + w = 0\)
• Equation 2: \(x + y + 2z + 2w = 0\)
• Equation 3: \(2x + 2y + 3z + 4w = 1\)
• Equation 4: \(2x + 3y + 4z + 5w = 2\)

To solve a system like this, you can use various methods, including graphing, substitution, elimination, or matrix techniques. The goal is to find the values of the variables that satisfy all the equations simultaneously.
However, sometimes a system might not have a solution due to conflicting equations. When equations are inconsistent, it means there is no set of variable values that can satisfy all equations at once.

Matrix Row Reduction

Matrix row reduction is a technique to simplify a system of linear equations using matrices. By representing the system as a matrix, we can systematically eliminate variables using row operations until the solution, or lack of one, becomes apparent.
The original system is rewritten in matrix form as: \[\begin{bmatrix} 1 & 1 & 1 & 1 \1 & 1 & 2 & 2 \2 & 2 & 3 & 4 \2 & 3 & 4 & 5 \end{bmatrix}\begin{bmatrix} x \y \z \w \end{bmatrix} = \begin{bmatrix} 0 \0 \1 \2 \end{bmatrix}.\]

• First, subtracting row 1 from row 2, twice from row 3, and row 4 simplifies the initial matrix.
• Then, further operations continue reducing until the matrix reveals any inconsistencies.

With row reduction, the goal is to transform the matrix into a form where it becomes clear whether solutions exist or not.

Inconsistent Systems

An inconsistent system of linear equations is a system with no possible solutions. This happens when the equations contradict each other, meaning there's no point that satisfies all equations simultaneously.
When using row reduction, you might identify an inconsistency. For instance, if a row in the matrix forms something like \(0 = 1\), it highlights a contradiction.

• In our system, after row-reduction, a row implies \(0 = 1\), clearly a false statement.

This means the system doesn't intersect at any point in the variable space, confirming it's inconsistent. Understanding such outcomes is crucial, as it tells you to re-evaluate assumptions, data, or modelling if unintentional inconsistencies arise in practical scenarios.