Problem 34 - Algebra and Trigonometry

Question

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} y-5 z \qquad=7 \\ 3 x+2 y \qquad=12 \\ 3 x \qquad +10 z=80 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent; no solution exists.

Step 1: Identify the System of Equations

We are given the following system of linear equations: \( y - 5z = 7 \), \( 3x + 2y = 12 \), and \( 3x + 10z = 80 \). Our goal is to determine if this system is inconsistent or dependent.

Step 2: Solve for Variables in Two Equations

Start with the first and third equations to express one variable in terms of others. From the first equation, solve for \( y \): \( y = 5z + 7 \). Use this in the second and the third equations.

Step 3: Substitution into the Second Equation

Substitute \( y = 5z + 7 \) into the second equation: \( 3x + 2(5z + 7) = 12 \). Simplify to get \( 3x + 10z + 14 = 12 \). This reduces to \( 3x + 10z = -2 \).

Step 4: Analyze Simplified System

Compare the two derived equations: \( 3x + 10z = 80 \) and \( 3x + 10z = -2 \). These equations are inconsistent because they cannot both be true (they have the same left-hand side but different right-hand sides).

Step 5: Conclusion

Since the simplified system gives contradictory equations, the original system of equations is inconsistent. This means there is no solution that satisfies all the equations simultaneously.

Key Concepts

Inconsistent System

An inconsistent system refers to a group of equations that cannot be satisfied by a common set of variable values. When dealing with linear equations, this occurs if the lines or planes represented by the equations do not intersect at any point. Consider each of the individual equations as a line in a two-dimensional space or a plane in a three-dimensional space. If these lines or planes don't meet or overlap, it indicates inconsistency.

In the context of the given system, we have three equations:

• \( y - 5z = 7 \)
• \( 3x + 2y = 12 \)
• \( 3x + 10z = 80 \)

When compared, the two equations derived from substitutions were \( 3x + 10z = 80 \) and \( 3x + 10z = -2 \), highlighting a direct contradiction. Such a contradiction confirms that no single set of values for \( x, y, \) and \( z \) can satisfy all equations simultaneously, making the system inconsistent.

Dependent System

A dependent system of equations is one in which all equations describe the same line or plane, essentially making one or more of them redundant. This redundancy indicates that the equations are linearly dependent. Instead of having distinct solutions, a dependent system can have infinitely many solutions along a common line or plane.

In practical terms, this type of system results in overlapping equations when plotted graphically. An indicator of a dependent system can be equations that are scalar multiples of each other, meaning they are essentially the same equation.

In our exercise, if the system under analysis were dependent, simplifying the equations would lead to identical equations rather than contradictory ones. Since our original system demonstrated a direct contradiction (indicating inconsistency), it is not dependent.

Linear Algebra

Linear algebra is a branch of mathematics focused on vector spaces and linear mappings between them. It is a crucial area of study for understanding systems of linear equations, which are foundational in fields such as engineering, physics, computer science, and economics. In linear algebra, lines and planes represented by linear equations are studied to determine interactions such as intersections, parallelisms, or dependencies.

The tools of linear algebra, such as matrices and determinants, allow for elegant solutions to complex problems involving multiple equations. In particular, these tools help to determine whether a system is inconsistent, dependent, or independent, and whether it has a single solution, many solutions, or no solution at all.

For our problem, the process outlined uses substitution and comparison, which are elemental techniques within linear algebra for determining the nature of a system. By establishing the validity and relationship between equations, you leverage linear algebra concepts to solve and interpret systems of linear equations efficiently.