Problem 16 - Precalculus

Question

In Exercises \(9-26\), put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent. $$ \left\\{\begin{aligned} 2 x-y+z &=-1 \\ 4 x+3 y+5 z &=1 \\ 5 y+3 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent, meaning it has no solutions.

Step 1: Write the System of Equations

First, list the given system of linear equations: 1. \( 2x - y + z = -1 \) 2. \( 4x + 3y + 5z = 1 \) 3. \( 5y + 3z = 4 \). Our aim is to express this system in triangular form to facilitate easy solving.

Step 2: Eliminate x from Equation 2

Multiply Equation 1 by 2: \( 4x - 2y + 2z = -2 \). Subtract this result from Equation 2: \((4x + 3y + 5z) - (4x - 2y + 2z) = 1 - (-2) \). This simplifies to \( 5y + 3z = 3 \).

Step 3: Compare New Equations

Notice that the new Equation 2 obtained, \( 5y + 3z = 3 \), is distinct from Equation 3, \( 5y + 3z = 4 \). Since these two equations are inconsistent (same coefficients for \( y \) and \( z \) but different constants), the system does not have a solution.

Step 4: Classify the System

Given the inconsistency between the two equations for \( y \) and \( z \) resulting from step 3, the system of equations is classified as inconsistent. An inconsistent system does not have any solutions.

Key Concepts

Triangular Form

To solve a system of linear equations efficiently, we often aim to arrange it into a "triangular form". This format is akin to a triangular matrix in linear algebra. Here's what this means: the system is organized such that each equation has fewer variables than the one before it.

• The first equation should involve all variables. In our case, it was made up of terms like \(2x - y + z = -1\).
• The second equation should have one less variable, often eliminating the first variable. We achieve this by eliminating \(x\) through steps like multiplication and subtraction.
• The third, or last equation, involves the fewest variables, for example, just \(y\) and \(z\) in this exercise.

This setup systematically reduces complexity, showing how each variable, one by one, can potentially be solved, or helps in identifying inconsistencies, as seen in this exercise.

Consistent vs Inconsistent Systems

In linear algebra, a system of equations is either consistent or inconsistent. A consistent system has at least one solution, while an inconsistent system doesn’t have any.
To determine consistency, examine the resulting equations after applying methods like substitution or elimination:

• If any simplification leads to contradictory equations, such as \(5y + 3z = 4\) and \(5y + 3z = 3\), the system is inconsistent.
• A consistent system either has a unique solution \((consistent independent)\) or infinitely many solutions \((consistent dependent)\).

In our example, we discovered that different equations for the same variables unexpectedly resulted, revealing the system to be inconsistent.

Solving Linear Systems

Solving linear systems involves finding values for variables that satisfy all given equations concurrently. Here's a step-by-step strategy:
1. Convert the system to triangular form for easier management.
2. Apply elimination methods: use operations like addition, subtraction, or multiplication to zero out variables in successive equations.

• This example involved eliminating \(x\) from the second equation, forming a simpler two-variable equation.
• Compare and contrast these transformed equations to identify inconsistencies, if any.

By working systematically, you can decipher whether a solution exists or not. If a conclusion indicates contradiction, no solution is possible, as it was here.

Linear Equations Classification

When dealing with linear systems, classification is vital to predict the nature of solutions effectively. Linear systems fall into three distinct categories:

• Consistent Independent: A system with exactly one solution set, characterized by distinct parallel equations.
• Consistent Dependent: Infinitely many solutions where all equations describe the same line or plane.
• Inconsistent: No solution due to contradictory equations, as identified through operations like eliminating variables and comparison.

In practice, classifying systems guides us in choosing the right method for solving and understanding what to expect from our solutions, just as we experienced with our inconsistent outcome in this task.