Problem 18 - 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 \(z\). $$\begin{aligned} 3 x-2 y-8 z &=1 \\ 9 x-6 y-24 z &=-2 \\ x-y+z &=1 \end{aligned}$$

Step-by-Step Solution

Answer

The system is inconsistent and has no solutions.

Step 1: Observe the System of Equations

The given system of equations is: \(3x - 2y - 8z = 1\), \(9x - 6y - 24z = -2\), and \(x - y + z = 1\). We will first attempt to simplify this system and check for dependency among the equations.

Step 2: Simplify Second Equation

Divide the second equation, \(9x - 6y - 24z = -2\), by 3 to simplify it. We get \(3x - 2y - 8z = -\frac{2}{3}\).

Step 3: Compare Simplified Equations

Notice that the first equation \(3x - 2y - 8z = 1\) and the simplified second equation \(3x - 2y - 8z = -\frac{2}{3}\) are not identical, suggesting inconsistency. This indicates that the system has no solutions.

Step 4: Verify Equation Consistency

We verify if the system is inconsistent by substituting the coefficients into the first and simplified second equations. As these produce differing results for the same coefficient combination, it confirms that there is no solution to the system.

Key Concepts

Consistent and Inconsistent Systems

When dealing with systems of equations, understanding the nature of solutions involves examining if the system is consistent or inconsistent. A **consistent system** means that there is at least one solution that satisfies all the equations. On the other hand, an **inconsistent system** has no solution because the equations contradict each other, as observed in this set of equations.

To determine if a system is inconsistent, you can look for scenarios where simplified forms of the equations have the same left-hand side but different constants on the right-hand side. For example:

• The equations \(3x - 2y - 8z = 1\) and \(3x - 2y - 8z = -\frac{2}{3}\) have identical expressions on the left side but are equal to 1 and \(-\frac{2}{3}\) respectively on the right, clearly indicating a contradiction.

Such contradictions affirm the system's inconsistency, leading to the conclusion that there is no solution that satisfies all equations simultaneously.

Dependent Equations

In a system of equations, **dependent equations** are those that can be expressed as multiples or linear combinations of each other. This means if you have one equation, the others don't provide new information. They confirm or restate what's already known.

In a dependent system, all equations are essentially the same when simplified. If one equation is a scalar multiple of another, the system does not offer independent relationships, thus termed dependent. For instance:

• In our original problem, if \(9x - 6y - 24z = -2\) were to simplify exactly to \(3x - 2y - 8z = 1\), they would be dependent, but given they simplify to different constants, they are not dependent.

Recognizing dependence helps in simplifying systems to more manageable forms for solutions.

Analytical Methods

**Analytical methods** for solving systems of equations involve finding solutions by algebraic manipulation rather than graphing or numerical estimation. These methods are powerful tools in finding precise solutions and include techniques such as substitution, elimination, and matrix operations.

In learning to solve systems analytically, key strategies include:

• **Substitution**: Solving one equation for a variable and substituting that expression in other equations.
• **Elimination**: Adding or subtracting equations to cancel out a variable, gradually reducing the system to fewer unknowns.
• **Reduction of matrices**: Using row operations to bring an augmented matrix into row echelon form or reduced row echelon form to find solutions.

Analytical methods provide exact solutions when applicable, and understanding their use is fundamental in higher mathematics such as linear algebra and calculus.