Problem 37 - A Graphical Approach to College Algebra

Question

\(\begin{aligned} &x-4 y+2 z=-2\\\ &x+2 y-2 z=-3\\\ &x-y \quad=4 \end{aligned}\)

Step-by-Step Solution

Answer

No solution exists; the system is inconsistent.

Step 1: Identify Variables and Equations

We have three equations with three variables, namely, \(x\), \(y\), and \(z\). The equations are: 1. \(x - 4y + 2z = -2\)2. \(x + 2y - 2z = -3\)3. \(x - y = 4\). These are linear equations that can be solved simultaneously to find the values of \(x\), \(y\), and \(z\).

Step 2: Solve for x from Equation 3

From the third equation, \(x - y = 4\), we can solve for \(x\) in terms of \(y\): \(x = y + 4\).

Step 3: Substitute x in Equations 1 and 2

Substitute \(x = y + 4\) into Equations 1 and 2:- Equation 1 becomes \((y + 4) - 4y + 2z = -2\).- Equation 2 becomes \((y + 4) + 2y - 2z = -3\).

Step 4: Simplify Equations

Simplify the equations obtained from Step 3:- Equation 1 simplifies to \(-3y + 2z = -6\).- Equation 2 simplifies to \(3y - 2z = -7\).

Step 5: Add Simplified Equations

Add the simplified equations from Step 4 to eliminate \(z\):\[(-3y + 2z) + (3y - 2z) = -6 - 7\] This simplifies to \(0 = -13\), which indicates an inconsistency.

Step 6: Analyze Result

The equation resulting in \(0 = -13\) after adding indicates that these equations do not have a consistent solution. This is a contradiction, and therefore the system of equations is inconsistent, meaning no simultaneous solution exists.

Key Concepts

Inconsistent System

Imagine trying to solve a puzzle, but the pieces just don't fit together. That's what happens with an inconsistent system of linear equations. It means there is no set of values for the variables that can simultaneously satisfy all of the given equations. This is like trying to draw three parallel lines that intersect at a point. Quite impossible, right?

In the exercise, this inconsistency is highlighted in Step 5, where the pesky equation 0 equals -13 appears. This equation makes no mathematical sense, and serves as a bright red flag that the equations conflict with each other.

• Inconsistent systems often happen due to errors in equations or specific conditions where they inherently can't be solved together.
• They are identified when you progress through the math and end up with a mathematical contradiction.

But don't worry, recognizing an inconsistent system can be a breakthrough moment. It saves you the hassle of false conclusions from impossible equations!

Linear Equations

Linear equations are like the building blocks of mathematical relationships. They come in the form of expressions that plot straight lines when graphed. These lines help us understand how changes in one variable affect another linearly.

Key features of linear equations include:

• The powers of the variables (like x, y, z) are always 1.
• They form straight lines when graphed.
• They can have one, some, or no solutions depending on how the lines intersect.

In our exercise, each equation deals with variables x, y, and z in a linear fashion. Identifying these equations is the first step to solving complex problems. Just like connecting dots to play an exciting game of join-the-dots, solving for these lines gives a clear picture of how each variable dances in response to others.

Solving Equations

Solving equations is a process of unraveling mysteries encoded in math problems. For linear equations, it often involves manipulating and substituting values to reach the truth about the relationships between variables.

In our exercise:

• We start by identifying the variables and structuring the given equations.
• We then isolate one of the variables (like finding x in terms of y in Step 2).
• Substitute back to simplify the remaining equations.
• This simplification helps in uncovering any inconsistencies or solutions.

It's a series of logical steps, akin to peeling an onion layer by layer until you reach the core, or the solution. But as shown by the exercise, sometimes these steps lead to contradictions (like 0 = -13), highlighting that no solution exists, and thus an inconsistent system is at play.