Problem 30 - Precalculus: Mathematics for Calculus

Question

Find the complete solution of the linear system, or show that it is inconsistent. \(\left\\{\begin{array}{c}-x+2 y+5 z=4 \\ x-2 z=0 \\ 4 x-2 y-11 z=2\end{array}\right.\)

Step-by-Step Solution

Answer

The system is inconsistent with no solutions.

Step 1: Write the system of equations

The system of equations is as follows:1. \(-x + 2y + 5z = 4\)2. \(x - 2z = 0\)3. \(4x - 2y - 11z = 2\).

Step 2: Solve for one variable

From the second equation \(x - 2z = 0\), solve for \(x\):\[x = 2z\]

Step 3: Substitute into other equations

Substitute \(x = 2z\) into the first and third equations. The first equation becomes:\[-(2z) + 2y + 5z = 4\]\[2y + 3z = 4\]The third equation becomes:\[4(2z) - 2y - 11z = 2\]\[8z - 2y - 11z = 2\]\[-2y - 3z = 2\]

Step 4: Solve substitution results for remaining variables

Now solve the system:1. \(2y + 3z = 4\)2. \(-2y - 3z = 2\)Add the two equations to eliminate \(y\):\[0 = 6\]This results in a contradiction, implying there is no solution.

Key Concepts

Inconsistent Systems

An inconsistent system of equations is when there are no solutions that satisfy all equations simultaneously. In the given problem, after simplifying the equations through substitution, we reach the conclusion \(0 = 6\). This is a contradiction, meaning it is impossible to find a value that satisfies this equality. In other words, no combination of values for the variables \(x\), \(y\), and \(z\) will satisfy all the original equations in the system.

For linear systems, the inconsistency typically arises when the equations represent parallel lines or planes that do not intersect. In the context of three variables, this can extend to planes in 3D space that never meet. Recognizing an inconsistent system early can save time and effort when attempting to find a solution.

Key indicators of inconsistency include:

• Reaching a logical contradiction like \(0 = 6\).
• Identifying parallel equations that can't meet.

Recognizing these features helps ensure comprehension of when and why a system lacks solutions.

System of Equations

A system of equations, particularly a linear system, consists of multiple equations that involve the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously.

In this problem, the system is expressed as:

• \(-x + 2y + 5z = 4\)
• \(x - 2z = 0\)
• \(4x - 2y - 11z = 2\)

These are three equations involving the variables \(x\), \(y\), and \(z\). Linear systems can be solved using various methods, including substitution, elimination, and matrix operations.

Understanding a system of equations can sometimes reveal relationships or constraints between variables directly from the equations themselves. Each equation can be viewed as a constraint on the possible values of the variables. The intersection of these constraints potentially leads to a solution.

Substitution Method

The substitution method is a common technique for solving systems of equations, where you solve one equation for one variable and then substitute this expression into the other equations. This approach aims to simplify the system by reducing the number of variables step by step.

In our exercise, we first solve the second equation \(x - 2z = 0\) for \(x\):\(x = 2z\). We then substitute this expression into the other equations, transforming them to only involve \(y\) and \(z\).

This results in a simpler system:

• \(2y + 3z = 4\)
• \(-2y - 3z = 2\)

By using substitution, you effectively reduce the number of equations and variables. However, as seen in this example, the simplification can lead to identifying situations like contradictions quickly, indicating the system is inconsistent and has no solution.

The substitution method is highly effective for smaller systems and when one equation can be easily solved for a single variable.