Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x+2 y+3 z &=8 \\ 3 x-y+2 z &=5 \\ -2 x-4 y-6 z &=5 \end{aligned}$$

Step-by-Step Solution

Answer

The system is inconsistent; no solution exists.

Step 1: Analyze the given equations

We start by examining the system of equations to check for dependency and consistency. The three equations are:\[ \begin{aligned} x + 2y + 3z &= 8 \ 3x - y + 2z &= 5 \ -2x - 4y - 6z &= 5 \end{aligned} \]

Step 2: Determine dependency

Two equations are dependent if one can be written as a multiple of the other. Let's check if the third equation is a multiple of the first:\[ (-2)(x + 2y + 3z) = -2x - 4y - 6z \] This equals the left side of the third equation but with a right side of \(-16\), not 5. Thus, these are not multiples of each other, indicating the equations are not all dependent.

Step 3: Use elimination method for unknowns

We will eliminate one of the variables from the first two equations. Multiply the first equation by 3 and subtract the second equation: \[ \begin{aligned} 3(x + 2y + 3z) &= 24 \ 3x - y + 2z &= 5 \end{aligned} \] Subtracting these gives: \[ \begin{aligned} 6y + 7z &= 19 \end{aligned} \]

Step 4: Eliminate variables from remaining equations

From Step 1, try to eliminate the variable x from the first and third equations:Multiply the first equation by 2: \[ 2(x + 2y + 3z) = 16 \] Subtract from the third equation: \[ -2x - 4y - 6z = 5 \] Subtracting gives us:\[ 0 = -11 \], leading to inconsistency.

Step 5: Conclude inconsistency

Since the resulting equation \(0 = -11\) is false, the system of equations is inconsistent and no solution exists. The equations do not intersect at any point.

Key Concepts

Dependent Equations

In the context of systems of equations, dependent equations are those where one can be derived from the other through multiplication or addition/subtraction methods. These equations essentially provide the same information, just in different forms.
For example, if you have two equations, where one equation is a multiple or a combination of the other, then these equations cover the same solution set. But, how do we identify dependent equations?

• Look for a possibility of simplifying one equation as a multiple of another.
• Cross-check by ensuring both the left-hand and right-hand sides match after simplification.
• For extended systems, similar principles apply: derive one equation as a linear combination of the others.

In the original exercise, despite checking for dependency, the third equation was not a suitable multiple of the first. Therefore, the equations were not dependent, which led us to further analyze using other methods.

Elimination Method

The elimination method is a systematic approach for solving systems of linear equations. The main goal is to eliminate one variable at a time, simplifying the system to a more easily solvable form.
Here’s how it works:

• Start by aligning the equations neatly in a way that similar terms are aligned vertically.
• Multiply one or more equations by constants to align coefficients of one variable.
• Subtract or add these new equations to eliminate one variable.
• Repeat the process with the new simpler equations to solve for another variable.

In the original solution, the elimination method was used to simplify equations and solve for a dependent system. The goal was to reduce the three original equations by removing variables systematically. However, through the process, it was revealed that inconsistency exists among the equations.

Inconsistent System

An inconsistent system occurs when there are contradictions in the equations such that no solution satisfies all the equations simultaneously. This often becomes evident during the process of elimination or substitution when you end up with a false statement, such as 0 = -11.
Here’s how to detect an inconsistent system:

• Use methods like elimination or substitution and observe if a contradiction arises.
• The outcome might look like '0 = 5', or '3 = -7', indicating no feasible solution.
• Graphical interpretation shows lines or planes that do not intersect anywhere.

In the given exercise, after applying the elimination method, an equation resulted in a contradiction of 0 = -11, clearly showing that the system has no solution and is thus inconsistent. Such systems indicate that the lines or planes represented by the equations never meet at a common point.