Problem 17 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{aligned} &\begin{array}{l} x-4 y+2 z=-2 \\ x+2 y-2 z=-3 \end{array}\\\ &x-y \quad \quad \quad=4 \end{aligned} $$

Step-by-Step Solution

Answer

The system has no solution, as it leads to a contradiction.

Step 1: Organize the system of equations

We have the following system of equations: \(x - 4y + 2z = -2\), \(x + 2y - 2z = -3\), and \(x - y = 4\). We'll arrange the equations for clarity before solving.

Step 2: Solve the third equation for x

Using the equation \(x - y = 4\), solve for \(x\). Add \(y\) to both sides: \(x = y + 4\). We will use this result to substitute for \(x\) in the other equations.

Step 3: Substitute \(x\) in the first equation

Substitute \(x = y + 4\) into the first equation: \((y + 4) - 4y + 2z = -2\). Simplify to \(-3y + 2z = -6\).

Step 4: Substitute \(x\) in the second equation

Similarly, substitute \(x = y + 4\) into the second equation: \((y + 4) + 2y - 2z = -3\). Simplify to \(3y - 2z = -7\).

Step 5: Solve the system formed by new equations

Now solve the system \(-3y + 2z = -6\) and \(3y - 2z = -7\). Add these equations to eliminate \(z\): \((-3y + 2z) + (3y - 2z) = -6 - 7\), leading to \(0 = -13\). This is a contradiction, indicating there is no solution.

Key Concepts

Substitution Method

The Substitution Method is a classic technique for solving systems of linear equations. It's often used when one of the equations can be easily solved for one variable, and then that expression is substituted into the remaining equations. This method helps to reduce a system of three variables into two, which simplifies the solving process.
The basic steps are:

• Solve one of the equations for one variable in terms of the others.
• Substitute the expression from the first step into the remaining equations.
• Solve the reduced system of equations.
• If necessary, substitute back to find the other variables.

In our exercise, the equation \(x - y = 4\) was solved for \(x\) to get \(x = y + 4\). By substituting \(x\) in the remaining equations, we transform the system, making it easier to handle.

Contradiction in Equations

A contradiction in equations arises when the process of solving a system results in a statement that is clearly false, such as \(0 = -13\), as seen in our exercise. This indicates that the original system of equations has conflicting or inconsistent relationships.
To identify a contradiction:

• Follow the steps of our solution method, like substitution or elimination, to reduce the system.
• Observe if the final resulting expression is a contradiction.

Whenever you encounter a contradiction, it signals an impossibility in the equations that corresponds to errors or misunderstandings in attempting to find a solution set that satisfies all the equations simultaneously.

No Solution in a System

When solving systems of linear equations, discovering that there is no solution means the equations in the system are inconsistent; they do not intersect at any point in their graphical representation. Such systems are referred to as inconsistent systems.
No solution typically means:

• The lines represented by the equations are parallel and never meet.
• There is a contradiction as explained in the previous section.

In our exercise, solving led to the statement \(0 = -13\). This indicates that there is no set of values for \(x\), \(y\), and \(z\) that can satisfy all the given equations simultaneously. When working with systems, understanding that a no solution outcome is possible can save valuable time and lead to deeper insights into the nature of linear equations.