Problem 27 - 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 with no solutions.

Step 1: Analyze the Given System

The given system of equations is as follows: \( x + 2y - z = 1 \), \( 2x + 3y - 4z = -3 \), and \( 3x + 6y - 3z = 4 \). These are three linear equations with three variables \( x \), \( y \), and \( z \). The goal is to find values of \( x \), \( y \), and \( z \) that satisfy all three equations.

Step 2: Simplify the System

Observe that the third equation, \( 3x + 6y - 3z = 4 \), can be simplified by dividing the entire equation by 3, resulting in \( x + 2y - z = \frac{4}{3} \). This is inconsistent with the first equation \( x + 2y - z = 1 \) since the left-hand side is identical, but the right-hand sides are different (\( 1 eq \frac{4}{3} \)).

Step 3: Check for Consistency

Having identified that two equations \( x + 2y - z = 1 \) and \( x + 2y - z = \frac{4}{3} \) have the same left-hand side but different results, the system is inconsistent. An inconsistent system has no solution because no set of \( x, y, z \) can satisfy both equations simultaneously.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, representing straight lines when graphed on a coordinate plane. These equations have terms with variables raised to the first power and often take the form of ax + by + cz + ... = d, where a, b, c, and d are constants. In this context, a linear equation's solution is any set of values for the variables that make the equation true.

When dealing with linear equations, you can expect the following characteristics:

• They depict linear relationships and have variables in first-degree power.
• They can be written in multiple forms like standard form, slope-intercept form, etc., depending on the number of variables involved.
• Simplifying and combining equations can help identify solutions or inconsistencies.

Grasping the base structure of linear equations is essential for solving systems involving multiple equations. Remember, the key is in understanding how these equations interact with one another.

Three Variables System

A three variables system involves linear equations with three distinct variables, such as x, y, and z. In these scenarios, you will encounter a system of equations that must be solved simultaneously to find a common solution.

Systems with three variables typically emerge in problems where spatial relationships, such as geometrical interpretations in three dimensions, are analyzed. Here are a few crucial points about these systems:

• You will have at least three equations to solve for the three unknowns.
• Each equation contributes to the overall picture of how the variables relate to each other.
• Solutions can sometimes represent a single point, a line, a plane, or even an empty set when no common solutions exist.

When tackling these systems, it's important to align equations accurately to spot patterns or inconsistencies, like identical left-hand sides with differing constant terms.

No Solution

The term "no solution" signifies an inconsistency within a system of linear equations. In simple terms, it means there's no possible set of values for the variables that can satisfy all the equations at the same time. This happens when equations contradict each other.

To identify a no solution scenario, certain signs can alert you:

• Two equations become identical in structure on one side but have differing results on the other side, much like parallel lines that never intersect.
• Simplification of one or more equations reveals contradictions in the possible values they could take.
• All possible operations lead to an absurdity, such as a nonzero value equaling zero.

When you encounter an inconsistency, such as parallel behaving equations, it indicates that in geometrical terms, they represent parallel planes that do not share a common intersection, leading to the conclusion of "no solution."