Problem 27 - Beginning and Intermediate Algebra

Question

Solve each system \(\begin{aligned} 6 x+3 y-3 z &=-1 \\ 10 x+5 y-5 z &=4 \\ x-3 y+4 z &=6 \end{aligned}\)

Step-by-Step Solution

Answer

The system is inconsistent (no solution).

Step 1: Check for consistency

Equation 1: \(6x + 3y - 3z = -1\)
Equation 2: \(10x + 5y - 5z = 4\)

Multiply Eq. 1 by \(\frac{5}{3}\): \(10x + 5y - 5z = -\frac{5}{3}\)

But Eq. 2 says \(10x + 5y - 5z = 4\).

Step 2: Conclude

Since \(-\frac{5}{3} \neq 4\), the first two equations are contradictory. The system is \(\textbf{inconsistent}\) and has no solution.

Key Concepts

Inconsistent System

An inconsistent system of equations is a set of equations that have no common solution. This typically occurs when the equations represent parallel lines, which means they will never intersect. Therefore, the system cannot have a point that satisfies all equations simultaneously.
In the given problem, after carrying out elimination, the equation we arrived at was of the form \(0x + 0y + 0z = -17\). This is a clear contradiction, as the left side equals zero while the right side does not. Such a result confirms the system is inconsistent, as no combination of \(x, y, z\) will satisfy the equation.
Remember, a system with no possible solution due to contradictory results is always inconsistent.

Elimination Method

The elimination method is a powerful tool for solving systems of linear equations. It involves adding or subtracting equations in such a way that one of the variables is eliminated, simplifying the problem to fewer variables.
Here's a practical guide on how it works:

• Choose a variable to eliminate from two equations.
• Adjust the coefficients of that variable by multiplying the equations suitably.
• Add or subtract the equations so that the chosen variable cancels out.
• Solve the simplified equation(s) for the remaining variables.

In the provided solution, this method was used to eliminate \(z\) by first adjusting the coefficients of \(z\) in equations 1 and 2. Multiplication by constants 5 and -3 ensured that adding the equations would cancel \(z\), leading to the simpler equation \(0 = -17\), identifying the system as inconsistent.

Linear Equations

Linear equations are polynomial equations of the first degree, meaning they involve variables raised only to the power of one. These equations often resemble straight lines when graphed on a coordinate plane, characterized by the general form \(Ax + By + Cz = D\).
Solving a system of linear equations involves finding values for the variables that satisfy all equations simultaneously. In contexts like the provided exercise, linear equations work together to potentially represent intersecting lines or planes, indicating a shared solution.
However, when such systems are inconsistent, as shown in this exercise, the parallel nature of the equations means they never share a solution point. Understanding the nature and behavior of linear equations is crucial for analyzing system consistency.

No Solution

A 'no solution' outcome in a system of equations means there are no values for the variables that will satisfy all equations simultaneously. This situation is common in inconsistent systems, as highlighted by the exercise.
The result \(0 = -17\) is a contradiction representing this "no solution" scenario. When graphing, this might translate to parallel lines or non-intersecting planes in higher dimensions.
If you find such a result, always check calculations to ensure accuracy. However, if the conclusion still stands, the system is confirmed to have no solutions due to its inherent inconsistency.