Problem 9 - Intermediate Algebra

Question

Solve each of the following systems. If the solution set is \(\varnothing\) or if it contains infinitely many solutions, then so indicate. $$ \left(\begin{array}{r} x-y+2 z=4 \\ 2 x-2 y+4 z=7 \\ 3 x-3 y+6 z=1 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is \(\varnothing\) because the system is inconsistent.

Step 1: Identifying the System

We are given a system of linear equations: \[\begin{align*}x - y + 2z &= 4 \2x - 2y + 4z &= 7 \3x - 3y + 6z &= 1\end{align*}\] We need to find the values of \(x\), \(y\), and \(z\) that satisfy all three equations.

Step 2: Check Linear Dependency

Observe that the second and third equations are multiples of the first equation. This suggests that the equations may be linearly dependent. Multiply the first equation by 2:\[2(x - y + 2z) = 2 imes 4 ightarrow 2x - 2y + 4z = 8\] Compare with the second equation:\[2x - 2y + 4z = 7\]The left-hand sides are identical, but the right-hand sides are different, indicating inconsistency.

Step 3: Conclusion about the Solution Set

Since the equations are linearly dependent and lead to contradictory statements (e.g., \( 8 = 7 \)), the system has no solutions.This means the solution set is empty, denoted as \(\varnothing\).

Key Concepts

Linear Dependency in Systems of Equations

When examining systems of linear equations, it is essential to determine whether the equations are linearly dependent. Linear dependency occurs when at least one equation in a system can be obtained from a linear combination of the others. This means:

• One equation can be derived by adding, subtracting, or scaling the others.
• If equations are linearly dependent, they describe the same geometrical plane in multi-dimensional space.

In our given system:\[\begin{align*}x - y + 2z &= 4 \2x - 2y + 4z &= 7 \3x - 3y + 6z &= 1\end{align*}\]we observe that by multiplying the first equation by some constant, we get the form of the second and third equations. However, if these scaled forms do not agree on all sides of the equation, it may suggest an issue such as inconsistency, as the rest of the system cannot hold together.

Understanding the Solution Set

The solution set of a system of linear equations is the collection of all possible solutions that satisfy all the equations in the system simultaneously. A solution set can be:

• Empty: This means there are no values for the variables that satisfy all equations, often due to linear dependency leading to inconsistencies.
• One unique solution: The system has a distinct solution for each variable.
• Infinitely many solutions: Typically occurs in systems with more equations than variables or with dependent equations that describe the same geometric figures.

In our problem, we established a contradiction after identifying the linear dependency, leading to an empty solution set. Thus, there are no solutions that can satisfy all three given equations simultaneously.

Identifying an Inconsistent System

An inconsistent system of equations is one where no solution exists. This typically occurs when the system is over-determined with dependent equations resulting in contradictory equations. Signs of inconsistency include:

• After transformations, identical left-hand sides with differing right-hand values, indicating a contradiction (e.g., equating 8 to 7).
• The geometric interpretation where described lines or planes in the system do not intersect at a common point, hence no common solution exists.

In our exercise, we found that after establishing linear dependency through scaling, the equations led to conflicting results – a clear symptom of an inconsistent system. Consequently, the problem's solution set is empty, symbolically represented by \( \varnothing \). This points to the inherent contradictions in the equations providing no universal solution.