Problem 63 - A Graphical Approach to College Algebra

Question

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} x-2 y+z &=5 \\ -2 x+4 y-2 z &=2 \\ 2 x+y-z &=2 \end{aligned}$$

Step-by-Step Solution

Answer

There's no solution as the system is inconsistent.

Step 1: Simplify the System of Equations

Start by observing the second equation, \(-2x + 4y - 2z = 2\). Notice that each term in this equation is divisible by \(-2\), so simplify by dividing the entire equation by \(-2\). This gives us: \(x - 2y + z = -1\).

Step 2: Identify Redundancies

Compare the first equation \(x - 2y + z = 5\) with the simplified second equation \(x - 2y + z = -1\). These equations represent parallel planes that never intersect because their right-hand sides are different. The system is, however, inconsistent. This implies there cannot be a consistent solution.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed in a coordinate system. They are characterized by variables raised to the first power and can include constants as well. Each equation represents a plane in the context of three variables (like the ones in our system).

For example, the equation \(x - 2y + z = 5\) is a linear equation because it involves three variables \(x, y,\) and \(z\) without higher powers or roots.
Linear equations are essential in systems of equations, where we aim to find common solutions that satisfy all the given equations.

In our exercise, identifying and simplifying linear equations allows us to determine relationships between variables. This process involves looking for patterns or common factors, as done in the solution when simplifying the second equation.

Inconsistent Systems

An inconsistent system occurs when there is no possible solution that satisfies all equations in the system simultaneously. This usually transpires when the equations represent parallel or identical lines (or planes in three-dimensional systems) that do not intersect.

In our problem, the comparison between the first equation \(x - 2y + z = 5\) and the simplified second equation \(x - 2y + z = -1\) reveals they are parallel planes. The same left-hand side and different constants indicate no intersection point, rendering the system inconsistent.

When you come across systems with equations that simplify to parallel lines or planes, it's crucial to understand that no single set of variable values can satisfy them all. Recognizing inconsistent systems allows us to conclude quickly and avoid unnecessary calculations.

Algebraic Solutions

Algebraic solutions involve using algebraic methods to solve systems of equations, seeking to find the values of the variables that satisfy all given expressions simultaneously.

The first step in algebraic solutions is often either substitution or elimination, which simplifies and reduces the system to a more manageable form. In our example, simplifying the second equation by dividing each term helps to clarify the inconsistency in the system.
Sometimes, you may express solutions in terms of a free variable if the system allows infinitely many solutions, unlike our inconsistent system which has none.

Working through algebraic solutions involves logical reasoning and simplification, focusing on altering equations systematically while preserving equality. This strategic approach helps in reaching conclusions about whether the system has one solution, infinitely many solutions, or, like in our case, no solution at all.