Problem 26 - 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. $$\left\\{\begin{array}{r}{x \qquad +3 z=3} \\ {2 x+y-2 z=5} \\ {-y+8 z=8}\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations is inconsistent.

Step 1: Analyze the System of Equations

We have a system of three equations with three variables:1. \( x + 3z = 3 \)2. \( 2x + y - 2z = 5 \)3. \( -y + 8z = 8 \). Our task is to identify whether this system is inconsistent (no solutions) or dependent (infinitely many solutions).

Step 2: Solve for One Variable in Terms of Others

We start by expressing \( y \) in terms of \( z \) from the third equation:\( -y + 8z = 8 \).Rearrange to get:\( y = 8z - 8 \).

Step 3: Substitute Into Other Equations

Use the expression for \( y \) from Step 2 in the second equation \( 2x + y - 2z = 5 \):Substituting \( y = 8z - 8 \), we have:\( 2x + (8z - 8) - 2z = 5 \).Simplify it to:\( 2x + 6z - 8 = 5 \),or, \( 2x + 6z = 13 \).

Step 4: Solve for x in Terms of z

Use Equation 1, \( x + 3z = 3 \), to express \( x \) in terms of \( z \):\( x = 3 - 3z \).

Step 5: Check for Consistency Among Equations

Using the expression derived in Step 4, \( x = 3 - 3z \), and substitute \( x \) in the revised equation from Step 3:\( 2x + 6z = 13 \).Replace \( x \) to get:\( 2(3 - 3z) + 6z = 13 \).Simplify:\( 6 - 6z + 6z = 13 \).This simplifies to \( 6 = 13 \), which is a contradiction. Hence, the system is inconsistent.

Key Concepts

Inconsistent System

When a system of linear equations is labeled as inconsistent, it means that there are no solutions that satisfy all the equations at once. In simpler terms, the lines or planes representing these equations do not intersect at any point.
A contradiction arises when you simplify the equations and end up with an equation that doesn't make sense, such as stating that 6 equals 13.
This is a clear indication that the system of equations cannot simultaneously hold true and thus has no solution. Identifying an inconsistent system promptly can save time and effort in problem-solving.

Dependent System

A dependent system is one where the equations are not entirely independent; instead, they are multiples or combinations of each other.
Such systems have infinitely many solutions because the lines or planes overlap entirely, creating the same set of solutions.
For instance, if you can manipulate one equation into another by multiplying or adding, that hints towards dependency.
Recognizing a dependent system involves simplifying the equations and observing that they reduce to equivalent forms.
In scenarios where the equations represent the same geometric entity, there's no unique solution, just a family of solutions.

Solving Linear Systems

Solving linear systems involves finding values of the variables that satisfy all equations simultaneously.
There are several methods available:

• Substitution, where one variable is expressed in terms of others and replaced in the remaining equations.
• Elimination, which involves adding or subtracting equations to eliminate one variable at a time.
• Graphing, which helps visualize solutions as intersections of lines or planes.

Each method can be chosen based on the specific problem setup or personal preference.
Understanding how to solve these systems is essential for analyzing the nature of the solutions, such as recognizing consistency, dependency, or inconsistency.