Problem 28 - Algebra and Trigonometry

Question

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+2 y-z &=1 \\ 2 x+3 y-4 z &=-3 \\ 3 x+6 y-3 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent because the third equation contradicts the multiple of the first equation.

Step 1: Inspect the Linear System

Given the system: \[\begin{align*} x+2y-z &= 1 \ 2x+3y-4z &= -3 \ 3x+6y-3z &= 4 \end{align*}\] We need to determine whether this system has a solution or is inconsistent. The system involves three equations and three variables \((x, y, z)\).

Step 2: Analyze the Third Equation

Notice that the third equation \(3x + 6y - 3z = 4\) is a multiple of the first equation \(x + 2y - z = 1\). Multiplying the entire first equation by 3, we get \(3(x + 2y - z) = 3 \times 1\), which results in \(3x + 6y - 3z = 3\), which is not equal to 4. This indicates inconsistency.

Step 3: Conclude Inconsistency

Since the third equation is supposed to be a multiple of the first equation but results in a different constant (3 instead of 4), this means there’s no set of \((x, y, z)\) that can satisfy all three equations simultaneously. Thus, the system is inconsistent.

Key Concepts

Systems of Equations

In mathematics, systems of equations refer to a set of equations with multiple variables. A solution to a system of equations must satisfy all equations in that system simultaneously. When dealing with such a system, you typically work with variables like

• x, y, and z

which you aim to find.

To solve a system, you can use several methods, such as substitution, elimination, or using matrices. These methods help in reducing the complexity and can lead to one of three outcomes:

• a unique solution
• infinitely many solutions
• no solution (inconsistent system)

For linear systems, such as the one given in the problem, where the equations are linear, these scenarios apply as well. Understanding how the equations interact with each other is critical in determining which of these is true for the system.

Inconsistent Systems

An inconsistent system of equations is one where no common solution exists for the equations. This means that there is not a single set of values for the variables that satisfy all the equations simultaneously.

In the given exercise, we identify the system as inconsistent by analyzing the relationships between the equations. Noticing multiples across equations can be a telltale sign of inconsistency.

Here's how:

• If one equation is a multiple of another but equates to a different constant, like the third equation being similar but unequal to the first, it shows conflicting requirements for the solution.
• This conflict signals that no point can lie on all planes (represented by equations) simultaneously.

Such conflicts lead to contradictions, confirming the system's inconsistency. Keeping an eye out for these inconsistencies is crucial when analyzing systems of equations.

Solutions of Equations

The solution to a system of equations is a set of values for the variables that makes each equation true. For instance, in a consistent linear system involving

• equations like x + 2y - z = 1
you'd find specific values of x, y, and z that work across the board.
• There can be exactly one solution, meaning the lines intersect at one point,
• or infinitely many solutions, where lines overlap completely.
In contrast, the given exercise shows an inconsistent system: the distinct constants in the equation, similar but not equal, indicate no intersecting point exists for all equations.

Thus, understanding solutions requires checking each possible resolution:
• Unique intersection
• Overlapping lines
• Non-intersecting lines
These checks determine whether the solution is viable or if the system is inconsistent. This thorough analysis is essential for deciphering solutions.